Lecture Notes for the course held by Rahul Pandharipande

Endrit Fejzullahu, Nikolas Kuhn, Vlad Margarint, Nicolas M¨uller,Samuel Stark, Lazar Todorovic July 28, 2014

Contents

0 References 1

1 Affine varieties 1

2 Morphisms of affine varieties 2

3 Projective varieties and morphisms 5 3.1 Morphisms of affine algebraic varieties ...... 7

4 Projective varieties and morphisms II 8

5 Veronese embedding 10 5.1 Linear maps and Linear Hypersurfaces in Pn ...... 11 5.2 Quadratic hypersurfaces ...... 11

6 Elliptic functions and cubic curves 12

7 Intersections of lines with curves 14

8 Products of varieties and the Segre embedding 14 8.1 Four lines in P3 ...... 16 9 Intersections of 17

10 The Grassmannian and the incidence correspondence 18 10.1 The incidence correspondence ...... 19

11 Irreducibility 20

12 Images of quasi-projective varieties under algebraic maps 20

13 Varieties defined by of equal degrees 20

14 Images of projective varieties under algebraic maps 21

I 15 Bezout’s Theorem 24 15.1 The resultant ...... 24

16 Pythagorean triples 25

17 The Riemann-Hurwitz formula 26

18 Points in 26

19 Rational functions. 28

20 Tangent Spaces I 28

21 Tangent Space II 31

22 Blow-up 31

23 Dimension I 32

24 Dimension II 32

25 Sheaves 32

26 Schemes I 33

0 References

Good references for the topics here:

1. Harris, Algebraic Geometry, Springer. 2. Hartshorne, Algebraic Geometry, Springer. Chapter I, but also beginning of Chapter II for schemes. 3. Shafarevich, Basic algebraic geometry I, II. 4. Atiyah, Macdonald (for basic commutative algebra).

1 Affine varieties

In this course we mainly consider algebraic varieties and schemes. It is worth noting that several definitions related to algebraic varieties are formally similar to those involving C∞- manifolds. (In algebraic geometry the local analysis of algebraic varieties is by commutative algebra, whereas in differential geometry the local analysis of C∞-manifolds is by calculus.) In the following we take the complex field C to be the underlying field (one could also consider any algebraically closed field instead; the discussion would be identical). n By affine n-space over C we mean C . For a subset X of C[x1, . . . , xn] we denote by V (X) := {x ∈ Cn | f(x) = 0 for all f ∈ X} the common zero locus of the elements of X. No- n tice that V (X) = V (hXi). A subset of C of the form V (X) for some subset X of C[x1, . . . , xn] is said to be an affine . By Hilbert’s theorem C[x1, . . . , xn] is Noethe- rian, so that every affine algebraic variety is in fact the common zero locus of finitely many polynomials. For a subset S of Cn the of S is the set of polynomials vanishing on S, I(S) := {f ∈ C[x1, . . . , xn] | f(x) = 0 for all x ∈ S}.

1 Both I and V are inclusion-reversing, and are√ connected by Hilbert’s Nullstellensatz: for ev- ery ideal I of C[x1, . . . , xn] we have I(V (I)) = I. Hence I and V induce a bijective inclusion- reversing correspondence between affine algebraic varieties and radical ideals of C[x1, . . . , xn]. The on Cn is defined by letting the closed subsets of Cn be the affine alge- braic varieties. The Zariski topology on a variety V in Cn is the induced (subspace) topology; a more intrinsic definition relies on the following notion: a subvariety W of V is a variety W in Cn such that W ⊂ V . Then a subset U of V is open in the Zariski topology of V if and only if V \ U is a subvariety of V . Let us now consider the sets {Uf } defined by f∈C[x1,...,xn] Uf := {x ∈ V | f(x) 6= 0}. Remark 1.1 {Uf } is a basis of the Zariski topology: let U ⊂ V be an open subset. f∈C[x1,...,xn] c By definition, U = V (I) for some ideal I of C[x1, . . . , xn]. Since C[x1, . . . , xn] is noetherian, I is finitely generated , i.e. I = hf1, . . . , fki for some polynomials. Therefore we have

c c k c k U = V (I) = V (hf1, . . . , fki) = (∩i=1V (hfii)) = ∪i=1Ufi , so every open subset is a union of just finitely many basic open subsets. It follows from this also that Cn is quasi-compact. Example 1.2 As every nonzero f ∈ C[x] has at most finitely many zeros, the Zariski topology on the affine line C1 is exactly the cofinite topology. Notice that in this topology every injection f : C → C is continuous, in contrast to the standard (euclidean) topology.

2 Morphisms of affine varieties

Let V ⊂ Cn an affine algebraic variety, U ⊂ V an open subset.

Definition 2.1 f : U → C is an algebraic (regular) function if for each p ∈ C there exist an open subset Wp ⊂ U containing p and gp, hp ∈ C[x1, . . . , xn] such that

1.0 6∈ gp(Wp). 2. f| = hp . Wp gp

Theorem 2.2 A map f : Cn → C is algebraic if and only if f is a polynomial function on Cn.

Proof. The “if” part is trivial. For the “only if” part, let f : Cn → C be an algebraic map. Then for every p ∈ C there is an open neighborhood W of p in n and polynomial functions h and g such that f| = hp and p C p p Wp gp gp never vanishes on Wp. Inside Wp we can find a basic open set p ∈ Urp ⊂ Wp, where

n Urp = {ρ ∈ C | rp(ρ) 6= 0}. It follows that for all points p ∈ we have a r ∈ [x , . . . , x ] such that f| = hp and C p C 1 n Urp gp n ∀q ∈ C : gp(q) = 0 → rp(q) = 0. p k With Hilbert’s Nullstellensatz it follows that rp ∈ (gp) and therefore rp = αpgp for some

αp ∈ C[x1, . . . , xn]. On Urp we have that rp, gp, αp are never 0.

On Urp we have h α h α f| = p p = p p Urp k gpαp rp

2 k n Define hˆ = h α andr ˆ = r . Then U = U k , and for every p ∈ there is r with r (p) 6= 0 p p p p p rp rp C p p and f| = hp (removing the hats again). Urp rp It follows that finitely many U suffice to cover n. Take p , . . . , p such that U ,...,U rp C 1 m rp1 rpm m cover C and write ri instead of rpi . We have (because of the cover) (r1, . . . , rm) = (1) and therefore m X ∃s1, . . . , sm ∈ C[x1, . . . , xn]: siri = 1 i=1

hi We have f|U = and Ur ∩ Ur = Ur r . On Ur r we have ri ri i j i j i j

hi hj = , hirj = hjri, hirj − hjri = 0 on Urirj ri rj n ri rj (hirj − hjri) = 0 everywhere on C (∗) |{z} |{z} 6=0 6=0

hirj − hjri = 0 ∈ C[x1, . . . , xn] We have to prove f ∈ C[x1, . . . , xn]. Let m X F = hksk k=1 F is a polynomial. We want to show F = f everywhere. m m m X X X riF = rihksk = rkhisk = hi skrk = hi, k=1 k=1 k=1 because of (∗). And on Uri we have h h f| = i ,F | = i Uri Uri ri ri

If V ⊂ Cn is an affine algebraic variety and f : V → C is an algebraic function, f is restriction of a polynomial (see exercises). If f1, f2 : V → C are algebraic and the same function ⇔ f1 − f2 ∈ I(V ). We denote by Γ(V ) the of algebraic functions V → C.

Proposition 2.3 [x , . . . , x ] Γ(V ) ∼= C 1 n I(V )

Proof. By f ∈ C[x1, . . . , xn] 7→ f|V : V → C we have [x , . . . , x ] C 1 n ⊂ Γ(V ) I(V ) So, to prove the proposition, we need to show that this is surjective. Go through the proof of Theorem 2.2 step by step. It is the same, except for one step. hi hj ˆ 2 Again, we have Ur ∩ Ur = Ur r . On Ur r we have = . Let hi = hiri, rˆi = r . And we i j i j i j ri rj i have hirj − rihj = 0 on Urirj . And

rirj(hirj − rihj) = 0 everywhere on V (see exercise 1.3 on the exercise sheets)

3 Example 2.4 V = (x2 − y3) ⊂ C2. We look at f : V → C ( x (x, y) 6= (0, 0) (x, y) 7→ y 0 (x, y) = (0, 0)

x Let U = V \{(0, 0)}. f|U = y . y is never 0 in U. Claim: f is not algebraic on V .

Proof. Suppose it were. Then there is a basic open set Ur, r ∈ C[x, y], with (0, 0) ∈ Ur, and there are h, g ∈ C[x, y] such that x h ∀(x, y) ∈ U \{(0, 0)} : = r y g and g is never 0 on Ur. We have gx − hy = 0 on Ur and r(gx − hy) = 0 on V . We have x2 − y3|r(gx − hy) because (x2 − y3) is a radical ideal. x2 − y3 can’t divide r since r(0, 0) 6= 0. Therefore x2 − y3|gx − hy Because g(0, 0) 6= 0, g has a nonzero constant term. Use the

C[x, y] → C[t] x 7→ t3 y 7→ t2 Then x2 − y3 7→ 0, but g(t3, t2)t3 − h(t3, t2)t2 6= 0 because the t3 term (which exists, because g has a constant term) cannot cancel. Therefore x2 − y3 - gx − hy and f is not algebraic. Let V,W ⊂ Cn affine algebraic varieties. Let U ⊂ V ⊂ Cn, Uˆ ⊂ W ⊂ Cm be Zariski-open subsets of the affine algebraic varieties.

Definition 2.5 Φ: U → Uˆ is algebraic (regular) if

1. Φ is continuous (with regard to the Z-top.) 2. ∀S ⊂ Uˆ and ∀f : S → C algebraic the composition open

−1 Φ f Φ (S) −→ S −→ C is algebraic.

Let Φ be an algebraic map. Then there is a correspondence [y , . . . , y ] [x , . . . , x ] C 1 n = Γ(W ) → Γ(V ) = C 1 n I(W ) I(V ) f 7→ f ◦ Φ

4 3 Projective varieties and morphisms

Definition 3.1 We define projective n-space over C as Pn := (Cn+1 \{0})/ ∼, where the equivalence relation ∼ is defined by x ∼ y iff there is λ ∈ C∗ with y = λx.

Remark 3.2 1. Pn is not a ; it is the set of one-dimensional subspace of Cn+1 (lines through the origin). n n+1 2. If x ∈ P is represented by (x0, . . . , xn) ∈ C , then (x0, . . . , xn) is a set of homogenous coordinates for x and denoted [x0, . . . , xn] (unique only up to multiplication by a scalar), ∗ for any λ ∈ C we have [λx0, . . . , λxn] = [x0, . . . , xn].

It is evident that polynomials f ∈ C[x0, . . . , xn] do not induce well-defined functions on complex projective n-space; there are however certain polynomials which allow meaningful discussion of their zero sets, also called projective algebraic varieties. In addition, we can define the Zariski topology for projective varieties. The closed sets of this topology are defined to be the projective algebraic varieties. First, we need the notion of an homogeneous ideal of the ring C[z0, . . . , zn]. The motivation for this lies in the fact that our objects can only be described by homogeneous polynomials, and homogeneous ideals allow us to decompose any element into its homogeneous parts:

Definition 3.3 An ideal I ⊂ C[z0, . . . , zn] is homogeneous if for any f ∈ I all the homogeneous components of f are again elements of I.

Example 3.4 Consider f = f0 + f1 + ... where fi are the homogeneous components of degree 2 i. Take for example: f = 9 + 13z0 + 3z1. In particular, f has a set of homogeneous generators. We have the following results concerning the homogeneous ideal:

Lemma 3.5 Let I ⊂ C[z0, . . . , zn] be homogeneous, then I has finitely many homogeneous generators.

Proof. Let us consider I ⊂ C[z0, z1, z2, . . . , zn]. Using Hilbert’s basis theorem, we have that I is finitely generated in C[z0, z1, z2, ...zn] (say I = (g1, g2, g3, . . . , gn)). Taking the homogeneous components of the generators we obtain the conclusion (I = (g00, g01, . . . , g10, g11,... )).

Lemma 3.6 Consider an ideal I ⊂ C[z0, . . . , zn], which is generated by homogeneous elements. Then I is a homogeneous ideal.

Proof. Let (F1,F2,F3,... ) be the homogeneous generators of I. It follows that f ∈ I can be Pn expressed f = giFi, where Fi are homogeneous polynomials of degree di and gi are some i=1 P polynomials (not necessarily homogeneous). The degree d part of f is it equal to i GiFi where Gi represents the degree d − di part of gi.

√ Lemma 3.7 If I is a homogeneous ideal I ⊂ C[z0, . . . , zn] then I is homogeneous.

5 √ k Proof. Consider f ∈ I, then ∃k ∈ N such that f ∈ I. It follows that f = fl + fd (where k k k fd is the maximal degree term of the√ polynomial, so f = (fl + fd) can be written as f = k k k fd + other terms. Then, f − fd ∈ I and we can conclude (by induction). Given the previous results we can consider W ⊂ Pn be an projective algebraic variety: the zero-locus of the homogeneous polynomials F1,F2,F3,...,Fl. Then it follows that the ideal I = (F1,F2,F3,...,Fl) is a homogeneous ideal.

Theorem 3.8 (Projective Nullstellensatz) Let I√be an ideal in C[z0, . . . , zn] such that its zero locus V (I) in Pn is non-empty. Then I(V (I)) = I. √ Proof. As in the affine Nullstellensatz only I(V (I)) ⊂ I requires proof; thus let f ∈ I(V (I)). n If V (I) ⊂ P is defined by the homogeneous polynomials F1,F2,...,Fk, then these polynomials define an associated affine variety Vˆ (I) in Cn+1. (As it is defined by homogenous polynomials, Vˆ (I) is a cone in the sense that x ∈ Vˆ (I) implies λx ∈ Vˆ (I) for all λ ∈ C∗.) It is clear that f ∈ I(V (I)) vanishes on Vˆ (I) \{0}, for if x ∈ Vˆ (I) \{0} then [x] ∈ V (I). Now if ˆ 0 ∈ V (I) (which always holds, unless one of the Fi is constant and non-zero) we use that there ˆ is [x] ∈ V (I), giving f(0) = 0.√ Thus f vanishes on the whole of V (I); hence by the affine Nullstellensatz f ∈ I(Vˆ (I)) = I. Remark 3.9 The condition V (I) 6= ∅ is necessary, as can be seen by considering the maximal ˆ n+1 n ideal I := (z0, . . . , zn) with √V (I) = {0} in C and hence V (I) = ∅ in P . We have I(V (I)) = (1), which is distinct from I = I. We will now introduce the notion of algebraic functions and maps. Note that they are also known as regular functions and maps, respectively. Let W ⊂ Pn be a projective algebraic variety and U an open subset of W .

Definition 3.10 A function f : U → C is called an algebraic function if for every p ∈ U there is a neighborhood W of p in U and homogeneous polynomials of the same degree h, g ∈ C[z0, . . . , zn] with g 6= 0 on W and f|W = h/g.

Secondly, algebraic maps are defined as follows:

Definition 3.11 1. Suppose U is an open subset of either an affine algebraic variety or a projective algebraic variety. In this case, we call U a quasi-. 2. Let U, W be quasi-projective varieties. A map f : U → W is algebraic if:

i) f is continuous w.r.t. the Zariski topology. ii) f respects precomposition with algebraic functions, i.e. ∀Y ⊂ W open, ∀g : Y → C algebraic : g ◦ f : f −1(Y ) → C is an algebraic function.

One easily checks that any open or closed subset of a quasi-projective variety is again a quasi-projective variety. Remark 3.12 The reader should note the following basic facts about algebraic maps. The proofs are straightforward from the definitions and left as exercises.

6 1. If V is a quasi-projective variety, f1, . . . , fn ∈ Γ(V ) elements of the ring of algebraic functions on V , then

n V → C (or more generally: any affine algebraic variety) ξ 7→ (f1(ξ), . . . , fn(ξ)) is an algebraic morphism. n 2. If V ⊂ P is a quasi-projective variety, and if F0,...,Fm ∈ C[Z0, ..., Zn] are homogeneous polynomials of the same degree, which moreover do not have a common zero on V , then

m V → P ξ 7→ (F0(ξ),...,Fm(ξ)) defines an algebraic morphism. 3. If Φ : V → W is an algebraic map between quasi-projective varieties, V˜ ⊂ V a subvariety, then the restriction ˜ Φ|V˜ : V → W is algebraic. If W˜ ⊂ W is a subvariety that contains the image of Φ, then the map Φ: V → W˜ obtained by restricting the range of Φ is algebraic.

3.1 Morphisms of affine algebraic varieties The study of the morphisms between affine algebraic varieties can be done by understanding their ring of functions. Given two affine algebraic varieties V ⊂ Cn and W ⊂ Cm with their corresponding ring of functions Γ(V ) and Γ(W ) we have the following correspondence: If Φ : V → W is an algebraic morphism, then there exists a homomorphism of C-algebras (i.e. takes constants into constants): Φ∗ : Γ(W ) → Γ(V ) γ 7→ γ ◦ Φ

Conversely, if there is a homomorphism of the C-algebras f : Γ(W ) → Γ(V ), then this gives a unique algebraic morphism Φ : V → W such that Φ∗ = f. Let

m Φ: V → C ξ 7→ (f(y1)(ξ), . . . , f(ym)(ξ)) where y1, . . . , ym are the coordinate functions on W . We need to show that Im(Φ) ⊂ W . Let g ∈ I(W ) ⊂ C[Y1,...,Ym]. Then

g ◦ Φ = g(f(y1), . . . , f(ym)) = f(g(y1, . . . , ym)) = f(0) = 0 |{z} | {z } f homomorphism =0 This implies Im(Φ) ⊂ W . Φ is an algebraic morphism by remark 3.12 1. (see also Lemma 3.6 in Hartshorne). Let γ ∈ Γ(W ). Then

∗ Φ (γ)(ξ) = γ(f(y1)(ξ), . . . , f(ym)(ξ)) = f(γ(y1, . . . , ym))(ξ) |{z} f homomorphism and we have Φ∗ = f. In conclusion, Φ and Φ∗ encode the same data.

7 Example 3.13 • Let

2 Φ: C → C t 7→ (t2, t3) Then Im(Φ) = V (x3 − y2). This is an affine algebraic variety. • Let

2 2 Φ: C → C (x, y) 7→ (x, xy) Then Im(Φ)C = {(0, ρ)|φ 6= 0} and Im(Φ) is not a quasi-projective variety. It follows that the image of an algebraic map need not be a quasi-projective variety.

4 Projective varieties and morphisms II

Recall that we defined a quasi-projective variety as an open subset of either a projective or an affine variety. This makes sense, as we will see that any open subset of an affine variety is isomorphic to, i.e. can be viewed as an open subset of a projective variety. Let n ≥ 1. For 0 ≤ i ≤ n, we have basic open subsets

n Ui := UZi = { [ξ0, ξ1, . . . , ξn] ∈ P | ξi 6= 0} of projective space, and maps Φi,Ψi defined by: n Φi : C → Ui (x1, . . . , xn) 7→ [x1, . . . , xi−1, 1, xi+1, . . . , xn] and

n Ψi : Ui → C   ξ0 ξi−1 ξi+1 ξn [ξ0, ξ1, . . . , ξn] 7→ ,..., , ,..., ξi ξi ξi ξi which is well defined on Ui. Note that the Ui cover all of projective space. Claim 4.1 Φi and Ψi are isomorphisms between varieties, which are inverse to each other.

Proof. It is easy to check that the composition of Ψi with Φi in any order is the identity map. So one only needs to show that both are algebraic maps. For the sake simplicity, we consider only the case when i is zero and drop the index 0 when referring to Ψ0, Φ0. 1. To prove continuity, we verify that the preimages of closed sets remain closed under both maps.

Let first W ⊂ U0 be a closed subset defined by homogeneous polynomials F1,...,Fk ∈ −1 C[Z0,...,Zn]. Then Φ (W ) is the common zero set of the set of polynomials

(Fi(1,X1,...,Xn))i=1,...,k ∈ C[X0,...,Xn] (check this!), thus closed in Cn. n Conversely, if V ⊂ C is defined by (fi)i=1,...,k ∈ C[X1,...,Xn], then we can construct ho- Z1 Zn deg f mogeneous polynomials (Fi)i=1,...,k in n+1 variables, by Fi[Z0, ..., Zn] := fi( ,..., )Z . Z0 Z0 0 −1 Check that these define the preimage Ψ (V ) in U0 as their set of common zeros.

8 −1 2. Let f : U → C be any function, where U is open in U0. Let W := Φ (U). We want to show that f ◦ Φ: W → C is regular if and only if f is. Note that f ◦ Φ(x1, . . . , xn) = ξ1 ξn f([1, x1, . . . , xn]) and f([ξ0, . . . , ξn]) = f ◦ Φ( ,..., ). ξ0 ξ0

Assume f is regular. Let x = (x1, . . . , xn) ∈ W , ξ := Φ(x). By assumption, there is an ˜ H open neighborhood U of ξ, such that f|U˜ = G |U˜ , where H and G ∈ C[Z0,...,Zn] are homogeneous polynomials of the same degree, and G doesn’t vanish on U˜. Then f ◦ Φ equals H(1,X1,...,Xn) on the preimage of U˜, which is an open neighborhood of x. Noting G(1,X1,...,Xn) −1 ˜ that G(1,X1,...,Xn) is never zero for any point in Φ (U), this means that we have found a neighborhood of x where f ◦ Φ is equal to a rational function. This proves that Φ is algebraic. The other direction works similarly: Assume f ◦ Ψ−1 is algebraic. Let ξ ∈ U, x = Ψ(ξ). By assumption, there is an open neighborhood W˜ of x in W on which f ◦ Ψ−1 equals a h ˜ ˜ ˜ −1 ˜ rational function g , and g does not vanish on W . Now U := Φ(W ) = Ψ (W ) is an open neighborhood of ξ in W , and we have by the above remark:

h( Z1 ,..., Zn ) h( Z1 ,..., Zn )ZD Z0 Z0 Z0 Z0 0 f| ˜ = = , U g( Z1 ,..., Zn ) g( Z1 ,..., Zn )ZD Z0 Z0 Z0 Z0 0 which holds for any positive D. In particular if D ≥ max{deg h, deg g}, then both the denominator and the numerator are homogeneous polynomials of the same degree in Z0,Z1,...,Zn. As ξ was arbitrary, this proves that Ψ is algebraic.

From what we have proven we directly obtain

Lemma 4.2 Any projective variety can be covered by finitely many affine varieties.

Using our newfound knowledge, we now want to compute the algebraic functions on CPn.

Lemma 4.3 Any algebraic function f : CPn → C is constant.

n Proof. Let f : CP 7→ C be algebraic. For 1 ≤ i ≤ 1, the restriction of f to Ui gives an algebraic function

n C → C (1) (x1, . . . , xn) 7→ f(x1, . . . , xi−1, 1, xi+1, . . . , xn), (2) but we already know that such a function must equal a polynomial Pi ∈ C[X1,...,Xn]. Let   d Z0 Zi−1 Zi+1 Zn Qi(Z0,...,Zn) := Zi Pi ,..., , ,..., , Zi Zi Zi Zi where d is a positive integer that is strictly bigger than the maximal degree of Pi. Qi is homogeneous of degree d and vanishes outside Ui. By construction,

Qi(Z0,Z1,...,Zn) f|Ui = d , Zi

9 and in particular on the overlap regions:

Q0(Z0,...,Zn) Qi(Z0,...,Zn) f|U0∩Ui = d = d . Z0 Zi So the equation d d Zi Q0(Z0,...,Zn) = Z0 Qi(Z0,...,Zn) is fulfilled at every point in Cn+1, where the zero’th and i’th coordinate are not equal to zero. However, at all other points, we see that both sides of the equations are zero. This means, the two polynomials take the same values on all of Cn, which implies they are the same. By unique d factorization we get, for 0 ≤ i ≤ n, that Zi divides Qi, and both have degree d, which means the quotient Qi d =: ci, Zi n has to be a constant, and by the above equality, ci = c0 for any i. As the Ui cover CP , f is everywhere constant and equal to c0.

Corollary 4.4 CPn is not isomorphic to any affine algebraic variety.

Proof. An isomorphism between varieties induces a isomorphism of C-algebras between their rings of functions. The only affine varieties whose ring of functions equals C are the points. But CPn is not isomorphic to a point.

5 Veronese embedding

The discussion so far has been about quasi-projective varieties and their regular functions f : X → C as well as their regular maps p : X → Y . We are interested to know what the image of a regular map of a projective variety is.

Suppose we have a map Pn → A, where A is an affine variety. What are the possibilities for this map? The claim is that this map must be constant. To see this, assume that for two n points ξ1 6= ξ2 ∈ P we have f(ξ1) 6= f(ξ2). This means f(ξ1) and f(ξ2) differ in at least one coordinate xi. Define a map g : A → C by g(x) = xi. Then the composition gf : P → C consists of a regular non-constant map, which we have already proved does not exist.

n m Now consider regular maps P → P . They are given by [z0, . . . , zn] → [F0,...,Fm] where all Fi are homogeneous polynomials in C[z0, . . . , zn] of the same degree and moreover have no common zeros.

Definition 5.1 The Veronese embedding of degree d ∈ N is the map

n (d+n)−1 νd : P → P n i i [z , . . . , z ] 7→ [(z 0 ··· z n ) Pn ]. 0 n 0 n 0≤ik≤d, k=0 ik=d

10 Example 5.2 The map 1 2 P → P [x, y] 7→ [x2, xy, y2] is the Veronese of degree 2 mapping from P1. Example 5.3 The Veronese with n = 1, d = 3 1 3 P → P [x, y] 7→ [x3, x2y, xy2, y3] is also called . n+d ( n )−1 i0 in Let Xi0,...,xn be the variable in P that corresponds to the monomial z0 ··· zn in the Veronese map. Then, we have on the image of the Veronese map, for the multi-indices I = i0, . . . , in,J = j0, . . . , jn,K = k0, . . . , kn,L = l0, . . . , ln such that I + J = L + K:

XI · XJ − XK · XL = 0 (3)

Proposition 5.4 The image of the Veronese is defined by the quadratic equations (3).

Corollary 5.5 The image of each Veronese map is a projective algebraic variety.

5.1 Linear maps and Linear Hypersurfaces in Pn

Definition 5.6 A linear hypersurface in Pn is defined by one equation:

n X 0 = akzk. k=0

Remark 5.7 When we deal with one hypersurface, we can assume a0 = 0, by a change of coordinates.

∗ Definition 5.8 The projective linear is PGLn+1(C) = GLn+1(C)/C .

n Remark 5.9 PGLn+1(C) is the group of automorphisms of P . Because are unique only up to a factor λ ∈ C∗, matrices defining a linear map Pn → Pn also define the same map if they differ only by some factor λ ∈ C∗.

5.2 Quadratic hypersurfaces

Definition 5.10 A quadratic hypersurface or in projective space Pn is the zero locus of some F ∈ C[z0, . . . , zn] of the form

n n n X 2 X X F = akzk + ak,lzkzl. k=0 k=0 l=k+1

11 Remark 5.11 A quadric n n n X 2 X X F = akzk + 2ak,lzkzl k=0 k=0 l=k+1 can also be written using a symmetric matrix M:   a0 a0,1 ··· a0,n   z0 a0,1 a1 ··· a1,n z ··· z     .  F = 0 n  . .. .   .   . . .  zn a0,n ··· an−1,n an | {z } =:M

Definition 5.12 The rank of a quadric is the rank of the symmetric matrix M associated with it.

Proposition 5.13 A quadric is determined, up to PGLn+1(C) by its rank.

Proof. M be the symmetric matrix of a quadric. Then, there is a P ∈ PGLn+1(C) such that 1   ...      T −1 −1  1  (P ) MP =    0   .   ..  0 where the number of 1’s is equal to the rank of M. Let Q = {x|xT Mx = 0} be the quadric associated with M. Then

PQ = {P x|xT Mx = 0}     1  .    ..           T T −1 −1 T  1  = {x|x (P ) MP x = 0} = x x   x = 0   0     .     ..      0 

Something about the genus of cubic and quadratic surfaces is missing here.

6 Elliptic functions and cubic curves

There is an intimate connection between cubic equations and elliptic functions (doubly periodic meromorphic functions), in particular the Weierstrass ℘-function. Consider a meromorphic function f on C which is doubly-periodic with periods 1 and i; let Λ denote the lattice in C generated by 1 and i (which is, as a set, simply Z[i]), defining also Λ0 := Λ − {0}. (One could consider instead of i any element of the upper half plane.) It follows that we also have

12 f(z) = f(z+λ) for all λ ∈ Λ. It is evident that f is determined by its values on the fundamental parallelogram Γ := [0, 1) × [0, i). If f is an entire function, then f is bounded on the compact set Γ by continuity, hence on all of C so that f must be constant by Liouville’s theorem. If f has only a simple pole ξ in the interior of Γ, then the residue theorem gives (taking ∂Γ to be positively oriented) 1 Z f(z)dz = resξf. 2πi ∂Γ

The integral vanishes as opposite sides of ∂Γ cancel one another by periodicity; hence resξf = 0 meaning that ξ is a removable singularity. Thus the argument above applies and shows f to be constant. Therefore, if f is to be nonconstant it must have at least two poles in Γ. We now define the elliptic function ℘, which has double poles at the points of Λ. It is defined by the infinite series 1 X  1 1  ℘(z) := + − , z2 (z − ω)2 ω2 ω∈Λ0 which converges compactly for all z∈ / Λ and hence defines a meromorphic function on C. As we may differentiate termwise, the derivative is 2 X 2 X 1 ℘0(z) = − − = −2 . z3 (z − ω)3 (z − ω)3 ω∈Λ0 ω∈Λ As the sum is invariant under z 7→ z + λ (λ ∈ Λ), we have ℘0(z) = ℘0(z + λ) for all λ ∈ Λ, hence ℘0 is an elliptic function. From this it follows that ℘(z) − ℘(z + λ) is constant and in fact vanishes identically as ℘(−λ/2) − ℘(λ/2) = 0 (λ/2 ∈/ Λ is not a pole), since ℘ is even. Thus ℘ is indeed doubly-periodic with periods 1 and i. If f is an elliptic function with periods 1 and i (not vanishing on ∂Γ and without poles on ∂Γ), then the integral 1 Z f 0(z) dz 2πi ∂Γ f(z) vanishes by an argument as in the first paragraph. By the argument principle this integral is equal to the number of zeros of f minus the number of poles of f (both counted with multiplicity). Thus ℘ has equally many poles as it has zeros. The Laurent expansion of ℘ about the origin is

 ∞ !2  ∞ 1 X 1 X zk 1 1 X X 1 ℘(z) = + − = + c zn where c = (n + 1) z2 ω2 wk ω2  z2 n n ωn+2 ω∈Λ0 k=0 n=1 ω∈Λ0 vanishes for n odd. Thus this can also be written as ∞ 1 X X 1 ℘(z) = + (2k − 1)G z2k−2 where G := , z2 k k ω2k k=2 ω∈Λ0 is the Eisenstein series of weight 2k (at i), and we find 4 24G ℘0(z)2 = − 2 − 80G + ... z6 z2 3 4 36G 4℘3 = + 2 + 60G + ... ; z6 z2 3 defining g2 := 60G2 and g3 := 140G3 we see that the differential equation

02 3 ℘ = 4℘ − g2℘ − g3

13 0 2 3 holds, since ℘ (z) − 4℘(z) + 60G2℘(z) = −140G3 + ..., the right hand side being an elliptic function without poles (and is hence constant). 0 2 3 Thus the points of the form (℘(z), ℘ (z)) satisfy the cubic equation y = 4x − g2x − g3. In fact if we define φ : C → P2 by φ(z) = [1, ℘(z), ℘0(z)] we get an induced bijection between the 2 3 2 cubic curve defined by y = 4x − g2x − g3 (embedded into P ) and the commutative group C/Λ. The latter is homeomorphic to a torus, as can be seen by considering the construction of the torus from its fundamental polygon; notice that the geometric structure of a torus is intrinsically doubly periodic. The cubic curve is in fact nonsingular (the roots of the polynomial 3 4x − g2x − g3 are distinct are in fact given by ℘(1/2), ℘(i/2), ℘(1/2 + i/2)), hence an elliptic curve and as such has a group law (turning the bijection into a group isomorphism). Ahlfors’ complex analysis, chap. 7, was recommended during the lecture. It was also suggested to read the entire book.

7 Intersections of lines with curves

n Definition 7.1 Let α = (α0, . . . , αn), β = (β0, . . . , βn) ∈ C be linearly independent vectors. A set S ⊂ Pn of the form

1 S := {[α0s + β0t, α1s + β1t, . . . , αns + βnt]|[s, t] ∈ P } is called a line.

Let F ∈ C[X,Y,Z] be homogeneous of degree d. We want to find the intersection of the zeros of F with the a line given by (α0, β0, γ0), (α1, β1, γ1). This is given by

F (α0s + α1t, β0s + β1t, γ0s + γ1t) = 0 | {z } =Fˆ(s,t) Either Fˆ = 0 or ˆ 1 1 2 2 d d F = (λ0s + λ1t)(λ0s + λ1t) ··· (λ0s + λ1t)

8 Products of varieties and the Segre embedding

We would like to investigate the products of algebraic varieties. In the affine case, this is a very simple matter. For f1, . . . , fr ∈ C[x1, . . . , xn], g1, . . . , gs ∈ C[y1, . . . , ym], let A = n m V (hf1, . . . , fri) ⊂ C and B = V (hg1, . . . , gsi) ⊂ C be affine varieties. Then A × B is a n m n+m subset of C × C , which in turn is isomorphic to C . We can now consider the fi and gi as elements of C[x1, . . . , xn, y1, . . . , ym]. It is clear that A × B = V (hf1, . . . , fr, g1, . . . , gsi). In the projective case, the above approach does not work, as Pn × Pm is not isomorphic to Pn+m. However, we can define an embedding from Pn × Pm to a projective space.

Definition 8.1 The map

n m (n+1)(m+1)−1 σ : P × P → P [x0, x1, . . . , xn] × [y0, y1, . . . , ym] 7→ [x0y0, x0y1, . . . , xiyj, . . . xnym], where 0 ≤ i ≤ n and 0 ≤ j ≤ m, is called the Segre embedding. We give Pn × Pm the structure of a projective variety by identifying it with its image.

14 We will prove that our definitions make sense: Claim 8.2 σ is injective.

n m Proof. Let c = [c0,0, c0,1, . . . , ci,j, . . . cn,m] be an element of Im(σ). Let (a, b) ∈ P × P such that σ(a, b) = c. Without loss of generality, a0 = b0 = c0,0 = 1. This enforces bj = c0,j for all 0 ≤ j ≤ m and ai = ci,0, which uniquely determines a and b. Therefore, the Segre embedding is bijective onto its image.

Claim 8.3 The image of σ is a variety of P(n+1)(m+1)−1.

Proof. We denote the complex polynomials in (n+1)(m+1) variables by C[z0,0, z0,1, . . . , zi,j, . . . zn,m]. For i 6= k and j 6= l, let us define Pi,j,k,l := zi,jzk,l − zk,jzi,l and I, the ideal generated by the n m Pi,j,k,l. It is clear that σ(P × P ) ⊂ V (I). For the other inclusion, take a c ∈ V (I). Without loss of generality, c0,0 = 1. For any k, l 6= 0, the polynomial P0,0,k,l gives us ck,l = ck,0c0,l. By taking a0 = b0 = 1, ak = ck,0 and bl = c0,l, we get (a, b) such that σ(a, b) = c. Remark 8.4 1. By definition, the Segre embedding is algebraic. This, together with Claim 8.2, implies that the Segre embedding is an isomorphism onto its image. 2. It is also possible to define an algebraic structure on Pn × Pm by using bihomogeneous polynomials. One can check that this is equivalent to our definition. 3. We can also define the product of two quasi-projective varieties X and Y. It is not very hard to check that σ(X × Y ) again is a quasi-projective variety. 4. If X and Y are quasi-projective, it is useful to know that their product X × Y is a categorical product, i.e. the projections on the first resp. second coordinate are algebraic maps, and if we have algebraic maps

f : Z → X g : Z → Y

then the product map

(f, g): Z → X × Y z 7→ (f(z), g(z))

is also algebraic. 3 Example 8.5 Given four skew lines L1,L2,L3,L4 in P , how many lines intersect all of them? Consider the vector space V of homogeneous polynomials of degree two in the variables x, y, z, w. This is a 10 dimensional space and its projectivization parametrizes quadrics in P3. Pick three generic points on each of the lines L1,L2,L3. The condition for a quadric Q ∈ P(V ) to contain a point is a linear condition on V . As the lines are chosen general and there are three points on each line, there will be a 1-dimensional subspace of V of homogeneous polynomials vanishing at all the points and therefore a unique quadric Q that contains the chosen points (Check that the conditions are independent!). The restriction of the polynomial that defines Q, to the lines Li, i = 1, 2, 3 is a polynomial of degree 2 that vanishes at three distinct points, hence must be zero. This shows that Q contains the three lines L1,L2,L3. As the lines chosen are generic, Q is a smooth quadric and after a change of coordinates we may assume it is given by V (xw − yz), which is isomorphic to P1 × P1 via the Segre map. 1 1 The lines L1,L2,L3 are then given P × Pi for Pi ∈ P , i = 1, 2, 3 some points (This follows as they are distinct and of degree 1). Any line that meets the three lines L1,L2,L3 must meet the quadric Q in three distinct points and with the same argument as above is contained in Q. It 1 1 must hence given by MR = R × P for R ∈ P a point.

15 Consider now the fourth general line L4. L4 meets the quadric Q in precisely 2 points 1 1 (R1,S1), (R2,S2) ∈ P × P . This shows that there are precisely two lines (namely MR1 and

MR2 ) incident to all four lines L1,...,L4.

8.1 Four lines in P3 3 Given four skew (i.e. pairwise non-intersecting) lines L1,...,L4 in P , how many lines intersect all of them?

3 3 Lemma 8.6 Let L1,L2,L3 be skew lines in P and let K1,K2,K3 be skew lines in P . Then, there is a M ∈ PGL4(C) such that MLi = Ki for i = 1, 2, 3.

Proof. The ring of polynomials of P3 is C[X,Y,Z,W ]. Without loss of generality we may assume that K1 = Z(X,Y ),K2 = Z(Z,W ),K3 = Z(X − Z,Y − W ). 0 Because each line can be defined by two points that are on the line, we can find M ∈ PGL4(C) 0 0 0 such that M L1 = K1,M L2 = K2, because M can be defined by mapping four points in general linear position to four points in general linear position. 0 0 0 Now, consider M L3. This cannot lie in X = 0 or W = 0, because then M L3 and M L2 would 0 0 0 not be skew or M L3 and M L1 would not be skew. Therefore M L3 intersects X = 0 and 0 W = 0 in one point each. Let [0 : a : b : c] be the point where M L3 intersects X = 0 and 0 let [d : e : f : 0] be the point where M L3 intersects W = 0. K3 is defined by the points [0 : 1 : 1 : 1] and [1 : 1 : 1 : 0]. Now, we need some N ∈ PGL4(C) that leaves K1,K2 fixed and maps [0 : a : b : c] to [0 : 1 : 1 : 1] and [d : e : f : 0] to [1 : 1 : 1 : 0]. N can be written as a block matrix AB N = CD 0 0 where A, B, C, D are 2 × 2 matrices. To fix K1 we need that N[0 : 0 : z2 : z3] = [0 : 0 : z2 : z3]. This implies B = 0. Fixing K3, similarly, implies C = 0. To map [0 : a : b : c] to [0 : 1 : 1 : 1] and [d : e : f : 0] to [1 : 1 : 1 : 0] we can set

1/d 0 0 0   0 1/a 0 0  N =   .  0 0 1/f 0  0 0 0 1/c

Set M = NM 0.

3 Lemma 8.7 Let L1,L2,L3 be skew lines in P . Then, the family of lines which meets all three lines sweeps out a quadric in P3.

Proof. By the previous lemma we can assume that

L1 = Z(X,Y ),L2 = Z(Z,W ),L3 = Z(X − Z,Y − W )

Let Q = V (XW − YZ). Then Q contains L1,L2,L3. Also Q is the image of the Segre map from P1 × P1. We have that

1 1 1 L1 = σ({[0 : 1]} × P ),L2 = σ({[1 : 0]} × P ),L3 = σ({[1 : −1]} × P )

16 1 1 For any P ∈ P the line σ(P × P ) intersects L1,L2,L3 and so

1 1 [ 1 Q = σ(P × P ) = σ(P × {P }) P ∈P1

and therefore the lines which intersect all of L1,L2,L3 sweep out at least Q. Let L4 be a line that intersects all of L1,L2,L3. Let P1 ∈ L4 ∩ L1,P2 ∈ L4 ∩ L2,P3 ∈ L4 ∩ L3. Then XW − YZ is zero on P1,P2,P3, and therefore it is zero on all of L4 as it has degree 2.

9 Intersections of quadrics

3 Let Q1,Q2 ⊂ P be quadrics. We want to determine Q1 ∩ Q2. We use the Segre to cut out the first quadric:

1 1 3 S : P × P → P [x0, x1] × [y0, y1] 7→ [x0y0, x0y1, x1y0, x1y1] =: [z0, z1, z2, z3]

The image is the zeros of z0z3 − z1z2. Step 1 Choose coordinates, such that ∼ 1 1 Q1 = P × P |{z} Segre

−1 1 1 3 Step 2 Now consider S (Q2) ⊂ P × P . Q2 ⊂ P is defined by the equation

2 2 2 3 c1z0 + c2z1 + c3z2 + c4z3 + c5z0z1 + ··· + c10z2z3 =: F (z0, z1, z2, z3)

−1 What is S (Q2)? Let

G(x0, x1, y0, y1) = F (x0y0, x0y1, x1y0, x1y1)

1 1 −1 Let p ∈ P × P . Then p ∈ S (Q2) ⇔ G(p) = 0. But being zero for G doesn’t even make sense if it is (only) homogeneous. It needs to be bihomogeneous (i.e. homogeneous in each set of variables separately).

Definition 9.1 A polynomial p(x1, . . . , xn, y1, . . . , ym) is bihomogeneous of bidegree (a, b) if it is homogeneous of degree a in the x’s and homogeneous of degree b in the y’s.

3 Example 9.2 x0x1y0 is bihomogeneous with bidegree (2, 3). If G is bihomogeneous of bidegree (a, b), then whether G(p) = 0 is well defined for p ∈ P1 × P1. Let p = [x0, x1] × [y0, y1] = [λx0, λx1] × [µy0, µy1] Then G(p) is defined up to a factor of λa · µb.

Proposition 9.3 Let G be bihomogeneous. Z(G) ⊂ P1 × P1 is an algebraic subvariety.

Proof. Trivial.

17 We have that Q2 = F (z0, z1, z2, z3) is homogenenous of degree 2. Then we have G(x0, y1, y0, y1) = F (x0y0, x0y1, x1y0, x0y1) is bihomogeneous of degree (2, 2). We have projection maps π1, π2 : 1 1 1 1 P × P → P for the x and the y coordinates respectively. Look at π2(Z(G)) ⊂ P . What are 1 the fibers of Z(G)? For [ξ0, ξ1] ∈ P , G(x0, x1, ξ0, ξ1) has either one or two roots as a polynomial 1 in [x0, x1] ∈ P . Can G(x0, x1, y0, y1) be zero for all [x0, x1]? We have

2 2 G(x0, x1, y0, y1) = x0P1(ξ0, ξ1) + x0x1P2(ξ0, ξ1) + x1P3(ξ0, ξ1) where P1,P2,P3 ∈ C[y0, y1], homogeneous of degree 2. This can be zero for all [x0, x1] if and only if [ξ0, ξ1] is a common root of P1,P2,P3. How many points of π2(Z(G)) are double (ramification)? Look at

1 2 mG : Py → P

[y0, y1] 7→ [P1(y0, y1),P2(y0, y1),P3(y0, y1)]

2 2 2 Consider the quadratic equation P1x0 + P2x0x1 + P3x1 in P . We have a double zero if the 2 discriminant P2 − 4P1P3 is zero. this has 4 solutions in general, because it is an equation of degree 4 in P1. Something about the Euler characteristic is missing here!

10 The Grassmannian and the incidence correspondence

Quadrics in different dimensions

• In P1 we just have 2 points. • In P2 we have a conic, and P1 can be embedded as this conic with the Veronese. • In P3, P1 × P1 can be embedded as the quadric by the Segre. • We skip P4 for now. • In P5 Grassmann and Pl¨ucker thought about it and we will consider it now.

You can think about Pn as

n n+1 P = {L|L ⊂ C is a 1-dim subspace} Define the Grassmannian

n Gr(r, n) = {S|S ⊂ C is a r-dimensional subspace} Can we express Gr(r, n) as a quotient? Let M be the space of r × n matrices, and let U ⊂ M ∼ be the subset consisting of elements with rank r. We have U/Glr = Gr(r, n) (by the row space of the matrices). A matrix has rank < r if every r × r minor has det = 0. Let γ ⊂ {1, . . . , n} be a set with |γ| = r.

dγ : M → C dγ(ξ) 7→ the of the γ-minor of ξ

We have C U = {ξ ∈ M | ∀γ ⊂ {1, . . . , n} with |γ| = r : dγ(ξ) = 0} ⊂ M

18 This is Zariski closed. How can we think of Gr(r, n) as an algebraic projective variety? Let

(n)−1 P : Gr(r, n) → P r n C ⊃ S 7→ [zγ,... ].

Here, zγ is the γ-minor of a matrix ξS that has the vectors of a basis of S as columns. The ˆ basis is not unique, so ξS is determined only up to Glr. For ξS = gξS, g ∈ Glr we have ˆ ˆ (dγ(ξs)),... ) = (dγ(g · ξS),... ) = det(g)(dγ(ξS),... ) so [zγ,... ] is fixed up to a scalar! This fits perfectly with the homogeneous coordinates and the map is well-defined. It is called the Pl¨uckerembedding P. Example 10.1 Given an element ξS as

1 0 a b 0 1 c d we see that the Gr(2, 4) is, in fact, 4-dimensional. We want to compute the Pl¨ucker P. It is

P(S) = [z12, z13, z14, z23, z24, z34] = [1, c, d, −a, −b, ad − bc]

The image satisfies the first Pl¨ucker equation

z12z34 + z14z23 − z13z24 = 0

5 (4)−1 5 This is a rank 6 quadric in P = P 2 . We have a bijection of a full-rank quadric in P via the Pl¨ucker to Gr(2, 4).

10.1 The incidence correspondence

Fix r, n ∈ N.

Definition 10.2 The set

n−1 I = {(S, p)|S ∈ Gr(r, n), p ∈ P(S) ⊂ P } is called the incidence correspondence.

Theorem 10.3 I ⊂ Gr(r, n) × Pn−1 is an algebraic subvariety.

Proof. As Gr(r, n) is an projective algebraic variety by the Pl¨ucker embedding, it is covered by affine opens UJ where J ⊂ {1, . . . , n} with |J| = r. UJ corresponds to the r-dimensional n subspaces of C where the determinant of the J-th minor is nonzero. The elements of UJ are therefore given (after reordering) by matrices of the form   1 x1,1 . . . x1,n−r  .. . .   . . .  1 xr,1 . . . xr,n−r

19 ∼ r(n−r) Therefore UI = C . n−1 To show that I is Z-closed it suffices to show that I ∩UJ ×P is Z-closed for all possible indices J. It is sufficient to check this for J = {1, . . . , r}; for all other indices, the procedure is the same. r(n−r) We have the variables x11, x12, . . . , x1,n−r, . . . , xr,n−r in C and homogeneous coordinates [z1, . . . , zn]. We want to find equations F1(xij, zk),..., (where the x are ”honest” coordinates, r(n−r) n−1 and the z are homogeneous, projective coordinates) that cut out IJ ⊂ C × P . We get equations

F1(x, z) = zr+1 − x11z1 − x21z2 − · · · − xr,1zr

F2(x, z) = zr+2 − x12z1 − x22z2 − · · · − xr,2zr . .

Fn−r(x, z) = zn − x1,n−rz1 − · · · − xr,n−rzr

In conclusion I ⊂ Gr(r, n) × Pn−1 is an algebraic subvariety.

Proposition 10.4 Let V ⊂ P3 be a projective algebraic variety. Then the set of lines in P3 which meet V form an algebraic subvariety of Gr(2, 4).

3 −1 Proof. Let π1 : I → Gr(2, 4), π2 : I → P be the projection maps. Then π1π2 (V ) is the set 3 −1 −1 of lines in P which intersect V . π2 (V ) is closed. π1π2 (V ) is closed because the image of a projective variety under an algebraic map is closed (see theorem 14.3). Example 10.5 Example defining the Catalan numbers..

11 Irreducibility

Basically page 5 in Hartshorne.

12 Images of quasi-projective varieties under algebraic maps

Definition 12.1 Let X be a subset of a topological space. X is called locally closed if X = U∩V for U open and V closed. A constructible set is a finite union of locally closed subsets.

Theorem 12.2 Let X,Y be quasi-projective varieties. Let f : X → Y be algebraic. Then Im(f) is constructible.

13 Varieties defined by polynomials of equal degrees

n Let V ⊂ P be a projective algebraic variety. V = Z(F1,...,Fk), where Fi ∈ C[z0, . . . , zn] is homogeneous of degree di. Let d = max{d1, . . . , dn}. We want to find polynomials Gi all of degree d such that V = Z(G1,...,Gm). Construct them as follows: Let d−dj Gi,j := zi Fj

20 0 where 0 ≤ i ≤ n, 1 ≤ j ≤ k. Obviously V ⊂ Z({Gi,j : 0 ≤ i ≤ n, 1 ≤ j ≤ k}) =: V . Let p ∈ V 0. We know that

d−dj z0 Fj(p) = 0 . .

d−dj zn Fj(p) = 0, and it follows that Fj(p) = 0. ⇒ V = Z(G1,1,...,Gn,k).

Theorem 13.1 Let V ⊂ Pn be a projective algebraic variety, then we can find an embedding of V in PN such that the image is defined by quadratic equations.

n Proof. Let V ⊂ P be a projective variety. Assume that V is cut out by F1,...,Fk all of degree d. Let

n N νd : P → P d d [z0, . . . , zn] 7→ [z ,... , z ] 0 |{z} n all other monomials be the d-Veronese map. The image of Pn is cut out by quadratic equations. V ⊂ Pn is cut out n by {quadrics that cut out νd(P ), linear terms corresponding to F1,...,Fk}.

14 Images of projective varieties under algebraic maps

The following will be useful: Fact 14.1 If G ∈ C[Z0,...,Zn] is a homogeneous polynomial of degree d ≥ 1, then UG = {ξ ∈ Pn|G(ξ) 6= 0} is an affine open, i.e. isomorphic to a closed subset of affine space. n N Proof. We identify UG with its image U under the d’th Veronese embedding ν : P → P . Let V = Im(ν) Then G gives rise to a linear polynomial L ∈ C[Z0,...,ZN ] s.t. U = UL ∩ V . So U N is a closed subset of UL which, as L is linear, is isomorphic to C . Fact 14.2 Any quasi-projective variety Y can be covered by finitely many affine open sets. This N means that we can write Y = ∪i=1Ui, where for each i, Ui is an open subset of Y isomorphic to a closed subset of the affine space Cm.

Proof. There are n ≥ 1, F1,...,Fk,G1,...Gl ∈ C[Z0,...,Zn] homogeneous polynomials, s.t. n Y ⊂ P is the set of points p, s.t. p is a zero of every Fi but not of every Gj. If we restrict our attention to the open set UGi , we see that Y ∩ UGi = V (F1,...,Fk) ∩ UGi . So it is clearly an open set of Y and at the same time a closed subset of UGi which can be viewed as a closed subset of affine space by the previous fact.

Theorem 14.3 Let f : X → Y be an algebraic morphism between varieties, where X is projective and Y quasi-projective. Then

(A) Im(f) is Zariski-closed in Y . (B) for Z ⊂ X Zariski-closed, f(Z) is Zariski-closed in Y .

21 Remark 14.4 Note that it suffices to prove either (A) or (B), as the other part is then easily implied. Proof. We will prove Part (A) of the theorem. More precisely, we will prove first that the graph of f (defined below) is Zariski-closed in X ×Y , and second that the projection ΠY : X ×Y → Y is a closed map, whenever X is projec- tive, Y quasi-projective, which can be reduced to the proof of the special case X = Pn , Y = Cm.

The first part works for any algebraic morphismf : X → Y between varieties. Define the graph of f, Γ(f) := {(x, y) ∈ X × Y |f(x) = y} We want to show Γ(f) ⊂ X × Y is Zariski closed. (4) As Y is a quasi-projective variety, we can view it as a subset Y ⊂ Pn for some n. Let g : X → Pn be the composition of f and the embedding i : Y → Pn. Now it is clear that (4) will follow, if we can show instead n Γ(g) ⊂ X × P is Zariski closed. (5) This can further be simplified. Consider the map

n n n (g, id): X × P → P × P (x, y) 7→ (g(x), y)

This is algebraic by universal property of the product. Moreover, Γ(g) is just the preimage of the diagonal n n n ∆ = {(x, x) ∈ P × P |x ∈ P } under this map. Thus to prove (5) and thus finish the proof of (4), we have to prove that ∆ is Zariski-closed. By taking a look at the Segre embedding

n n N P × P → P ([x0, . . . , xn], [y0, . . . , yn]) 7→ [xiyj]1≤i,j≤n we see that the ∆ as a subset of PN is defined by the equations

Zij = Zji 0 ≤ i < j ≤ n together with the equations for the Segre embedding. The second part takes a bit more work. We will start with reformulating the problem several times. More precisely, we will show how any of the following statements implies the one above it, and then prove the last:

The projection ΠY : X × Y → Y is a closed map w.r.t. the Zariski topology. (6)

Claim 14.5

(i) (6) holds when X is a projective, Y a quasi-projective variety. (ii) (6) holds when X = Pn, Y a quasi-projective variety. (iii) (6) holds for X = Pn, Y = U, where U is a closed subset of affine space. (iv) (6) holds for X = Pn, Y = Cm

22 (ii)⇒(i): As X is projective, X ⊂ Pn is a closed subset for some n. But then X × Y is closed inside Pn × Y . Now given V ⊂ X × Y closed, it is also closed in Pn × Y . So by (ii) its image under projection is closed in Y .

(iii)⇒(ii): Cover Y with finitely many affine opens U1,...,UN . Now note that any subset of Y is closed if and only if for i = 1, . . . , n its intersection with Ui is closed in Ui. Also note −1 that ΠY (Ui) = X × Ui. Now let V ⊂ X × Y closed, W ⊂ Y its image under projection. Then V ∩ X × Ui is closed in X × Ui. By (iii), its image under ΠY , which is just W ∩ Ui is closed in Ui. This holds for any choice of i, so W is closed in Y . (iv)⇒(iii): trivial

What is left is the proof of (iv).

Proof. Let Z ⊂ Pn × Cm be a closed subset. Z is the zero locus of polynomials

F1,...Fk ∈ C[Z0,...,Zn,Y1,...,Ym],

where Fi is homogeneous, when viewed as a polynomial in Z1,...Zn , say of degree di. We m Want to find polynomial equations that describe ΠY (Z) in C . It is easily seen that

F1(Z0,...,Zn, ξ1, . . . , ξm) n . n ΠY (Z) = {(ξ1, . . . , ξm) ∈ C | . have a common zero in P } Fk(Z0,...,Zn, ξ1, . . . , ξm)

Let us take a step back, and reconsider the general question of when a set of homogeneous polynomials G1,...,Gl in the variables Z0,...Zn has a common zero in projective space and when the opposite holds. Of the following statements any one is in an obvious way (by definition or by some basic result) equivalent to the next one.

n 1. G1,...,Gl have no common zero in P . n+1 2. The common zero locus of G1,...Gl in C is empty or equal to (0,..., 0). ( p (1) 3. (G1,...,Gl) = (Z0,...,Zn) p 4. (G1,...Gl) ⊃ (Z0,...,Zn)

5. There is a d ≥ 1 such that (G1,...,Gl) contains all monomials in Z0,...,Zn of degree d. Now if we apply the equivalence of 1. and 5. to our situation we immediately get

m ΠY (Z) = {ξ ∈ C |∀d ≥ 1 : (F1(Z, ξ),...,Fk(Z, ξ)) does not contain all monomials of degree d.} ∞ \ m = {ξ ∈ C |(F1(Z, ξ),...,Fk(Z, ξ)) does not contain all monomials of degree d.} d=1 ∞ \ =: Wd. d=1 As any intersection of closed sets is closed, we can consider one d at a time. For fixed d, all mono- mials of degree d are contained in the ideal (F1(Z, ξ),...,Fk(Z, ξ)) if and only if they are con- tained in the subset of homogeneous elements of degree d, denoted by (F1(Z, ξ),...,Fk(Z, ξ))d, which is a vector space, and in particular a subspace of the space of homogeneous polynomials

23 of degree d, which is given by the monomial basis. So (F1(Z, ξ),...,Fk(Z, ξ)) does not contain all monomials of degree d, if and only if

(F1(Z, ξ),...,Fk(Z, ξ))d 6= C[Z0,...,Zn]d.

(F1(Z, ξ),...,Fk(Z, ξ))d is generated as a vector space by

α S := {Fi(Z, ξ)Z |0 ≤ i ≤ n : di ≤ d, |α| = d − di}

So the question is if these generate a space of dimension strictly lower than the number of monomials of degree d. By this is the case iff for the matrix M where the columns are the coordinates of elements of S w.r.t. the monomial basis, every size-d minor has zero determinant. Note that this gives a finite set of polynomial equations in the entries of M, and this gives us polynomials in ξ1, . . . , ξm, as every entry of M is a polynomial expression of m those. Wd is the zero locus of the resulting polynomials, thus closed in C .

15 Bezout’s Theorem

Last time we talked about B´ezout’stheorem, which in some sense is the analogue of the Fun- 2 damental Theorem of Algebra. In particular, given H1 and H2 in P where H1 is of degree d1 and H2 of degree d2, then H1 and H2 intersect d1d2 times, where points are counted with multiplicity. One problem with this statement is that H1 and H2 can intersect in infinitely many points, say for example H1 = H2 = (x0). But at least we can restrict to the simple case: if H1 ∩ H2 has finitely many points, then |H1 ∩ H2| = d1d2, with multiplicities counted. It may be the case that, at least locally, H1 and H2 intersect in two points, but after some parametrization they intersect in only one point. We want our definition of intersection to be independent of the parametrization, so we must somehow put some weight in each point to count for multiplicities. The way we calculate the weight of the point pi is by considering the local ring at pi and then taking the quotient by the equations of H1 and H2. The dimension of this quotient is equal to mpi and we get that X |H1 ∩ H2| = mpi .

pi∈H1∩H2

15.1 The resultant

Let p, q ∈ C[x, y]:

n n−1 n p = α0x + α1x y + ··· + αny m m−1 m q = β0x + β1x y + ··· + βmy

We want to find a polynomial R such that R(α0, . . . , αn, β0, . . . , βm) = 0 ⇔ p, q have a com- mon root. Let Vn be the vector space of homogeneous polymomials of degree n. We have n n−1 n n (x , x y, . . . , y ) as a basis and dim Vn = n + 1. Here, denote P = P(Vn). Let

1 n−1 m−1 n m f : P × P × P → P × P [ˆl, p,ˆ qˆ] 7→ [ˆlp,ˆ ˆlqˆ]

24 f is an algebraic map (check this!), and therefore Im(f) ⊂ Pn × Pm is Z-closed. Let

Lp,q : Vn−1 ⊗ Vm−1 → Vn+m−1

(k1, k2) 7→ k1q + k2p

If there are k1 6= 0, k2 6= 0 such that

k1q + k2p = 0, then q|k2p. This is equivalent to p and q having a common factor. It follows that if p and q have no common factor, then ker f = 0 and we have

p, q have no common factor ⇔ det(Lp,q) = 0

n n−1 n Definition 15.1 The matrix of Lp,q in the standard basis x , x y, . . . , y is the resultant.

It is given by   α0 α1 ...... αn 0 ... 0  .. .   0 α0 ...... αn . .   .  m rows  ......   . 0     0 ... 0 α0 ...... αn  Lp,q =   β0 β1 ...... βm 0 ... 0   . .   0 β ...... β .. .   0 m  n rows  ......   . . . . 0  0 ... 0 β0 ...... βm

16 Pythagorean triples

In the following, we consider the solutions of the equation x2 + y2 = z2. Over C this is a case which we have considered already, as the solutions form the conic V (x2 + y2 − z2) in P2. Thus we consider instead the solutions in Q; over Q we may assume that z = 1 after dividing by z2 (for z = 0 we only have the trivial solution). Geometrically, we want to determine the rational points on the unit circle x2 + y2 = 1. The geometric idea is rather simple; (1, 0) is such a point, taking a line with rational slope through (1, 0) we obtain another on the unit circle, namely the intersection of the line with the unit circle. (All rational points on the unit circle are obtained in this way, since for every such point there is a unique line joining it to (1, 0), which turns out to have rational slope.) Denoting the slope by s, the points on the line are of the form (1, 0)+t(1, s) = (1+t, ts). Plugging this into x2 +y2 = 1 yields t = −2/(1+s2), so that the sought point has coordinates

 2s  s2 − 1 2s  1 − 2/(1 + s2), − = , − . 1 + s2 s2 + 1 s2 + 1

The family of these points are all rational points on the unit circle. In P2 this point has homogeneous coordinates [s2 − 1, 2s, s2 + 1]. Now we consider x2 +y2 = z2 as a , considering only integral solutions; the simplest nontrivial solution being (3, 4, 5).

25 17 The Riemann-Hurwitz formula

Let X and Y be surfaces of genus g and h, respectively. Furthermore let f be a branched covering map of degree d (i.e. every non-ramification point in X has exactly d preimages) which is ramified in only finitely many points. We denote these points by b1, . . . , bR and assume that at reach ramification only two sheets come together.

Theorem 17.1 (Riemann-Hurwitz formula)

2 − 2g = d(2 − 2h) − R (7)

Proof. (sketch) We choose a triangulation of X such that b1, . . . , bR are corners of triangles. The preimage of this triangulation is (homeomorphic to) a triangulation of Y. Observe furthermore that all faces and edges have exactly d preimages under f. This also holds true for all vertices except the b1, . . . , br which have d − 1 preimages each. Therefore, we have χ(Y ) = dχ(X) − R (8) It is also known that χ(X) = 2 − 2h and likewise χ(Y ) = 2 − 2g, giving 2 − 2g = d(2 − 2h) − R (9)

This can be generalized for ramifications of arbitrary multiplicity. R is then equal to the sum of all multiplicities (greater than 1) minus the number of ramifications.

We can use the Riemann-Hurwitz formula to calculate the genus of varieties Example 17.2 Let F be a homogeneous polynomial in three variables. If F ([0, 0, 1]) 6= 0, we can canonically project V (F ) onto P1 using the projection map π([x, y, z]) := [x, y]. π is a branched covering of degree d because for fixed x and y, F (x, y, z) is a polynomial in z of degree d. We also know that the genus of P1 is 0, giving h = 0. However, it is hard in general to determine the number and multiplicities of the ramifications and therefore R.

18 Points in projective space

2 Let V = {p1, . . . , pr} ⊂ P .

Proposition 18.1 V can be cut out by degree r polynomials.

Proof. Let S = {F |F = L1 ...Lr, where Li is a line through pi} The polyomials in S all have degree r. We have that V ⊂ Z(S). Let q ∈ P2 \ V . Then there a line through every point in V that misses q. Therefore q∈ / Z(S). ⇒ V = Z(S).

Example 18.2 A set of r distinct points in P2 cannot necessarily be cut out by polyomials 2 of degree r − 1. For example let L ⊂ P be a line and let p1, . . . , pr be distinct points on the line. Let F be a polynomial of degree r − 1 which vanishes on p1, . . . , pr. Then F vanishes everywhere on L.

26 n Definition 18.3 The points p1, . . . , pk ∈ P are in general linear position if corresponding n+1 vectors v1, . . . , vk ∈ C are as linearly independent as possible. This means that for each subset S with |S| ≤ n + 1, dim span S = |S|.

Proposition 18.4 Let

1 d φ : P → P [s, t] 7→ [sd, sd−1t, . . . , td]

1 be the d-Veronese. Let {p0, . . . , pd} ⊂ P be d + 1 distinct points. Then {φ(p0), . . . , φ(pd)} are in general linear position.

Proof. See exercise 3 on sheet 4.

Definition 18.5 Let P0,...,Pn be a basis of the homogeneous polynomials of degree n. Then the map

1 n P → P [s, t] 7→ [P0(s, t),...,Pn(s, t)] is called a rational normal curve.

Example 18.6 The Veronese map from P1 is an example of a rational normal curve.

n Theorem 18.7 Let p1, . . . , pn+3 ∈ P be in general linear position. Then there is a unique rational normal curve in Pn passing through these points.

Proof. Choose a basis of Cn+1 such that

p1 = [1, 0,..., 0]

p2 = [0, 1,..., 0] . .

pn+1 = [0, 0,..., 1].

We have pn+2 = [α0, . . . , αn]. We have ∀i ∈ {0, . . . , n} : αi 6= 0, because else the points would not be in general linear position. We can rescale the basis vectors such that

pn+2 = [1, 1,..., 1].

We have pn+3 = [λ0, . . . , λn], where ∀i ∈ {0, . . . , n} : λi 6= 0 and i 6= j ⇒ λi 6= λj, because of the general linear position. 1 n 1 We are looking for a rational normal curve φ : P → P and points ξ1, . . . , ξn+3 ∈ P such that φ(ξi) = pi. Denote for i ∈ {1, . . . , n + 1} : ξi = [1, ξi]. Now φ(ξi) = [P1(ξi),...,Pn+1(ξi)] = [0,..., 0, 1, 0,..., 0] implies that (setting x = t/s)(x−ξi) | Pk for k 6= i and therefore Y Pi = αi (x − ξj) j6=i

27 From φ([0, 1]) = [1,..., 1] we can infer that all αi are equal (we can set them equal to 1). We have Y Y  1 1  φ([1, 0]) = [ ξi,..., ξi] = ,..., = [λ1, . . . , λn+1] ξ1 ξn+1 i6=1 i6=n+1 1 and therefore we fix ξi = which is possible, because λi 6= 0. λi

19 Rational functions. 20 Tangent Spaces I

In Differential Geometry, tangent spaces, at least for smooth submanifolds, arise very naturally. The tangent space at a single point is best described as the collection of possible starting directions one can take when travelling from that point along the manifold. We want to develop a similar notion for algebraic varieties. For this, we will start with the definition for affine varieties and build from that towards a more general formulation. Let X ⊂ Cn be an affine algebraic variety, p ∈ X

Definition 20.1 (First definition of Tangent Space)

n 1. We define Tp(C ) to be the complex n-dimensional vector space generated by the abstract basis ∂1, . . . , ∂n. Pn 2. For f ∈ C[X1,...,Xn], v = i=1 vi∂i ∈ Tp, let

n X ∂if (D f) := v (p) v p i ∂x i=1 i

n 3. Tp(X) := {v ∈ Tp(C ) | ∀f ∈ I :(Dvf)p = 0} is the tangent space of X at the point p.

Note that for (Dvf)p the usual properties of differentiation, such as linearity and the Leibniz rule, hold. We will also write Dvf if there is no ambiguity concerning the basepoint.

Lemma 20.2 Let f1, . . . , fk be generators of I(X). Then

n Tp(X) = {v ∈ Tp(C ) | For i = 1, . . . , k :(Dvfi)p = 0}

n Proof. Let v ∈ Tp(C ) s.t. Dvf1 = ... = Dvfk = 0. Then for g = h1f1 + ... + hkfk ∈ I(X) we have

n n X X (Dvg)p = (Dv hifi)p = (Dvhi)pfi(p) + hi(p)(Dv(fi))p = 0 i=1 i=1

As, for any i, by assumption Dvfi = 0 and by definition of I(X): fi(p) = 0. So v ∈ Tp(X). This proves one inclusion. The other one is obvious.

28 Wen can always assume that up to a translation p = (0,..., 0). Then Tp(X) is especially easy to compute: Let I = (f1, . . . , fk). We can write

2 f1 = α11X1 + α12X2 + ... + α1nXn + O(X ) 2 f2 = α21X1 + α22X2 + ... + α2nXn + O(X ) . . 2 fk = αk1X1 + αk2X2 + ... + αknXn + O(X )

Then the tangent space is given as the of the Matrix   α11 α12 . . . α1n α α . . . α   21 22 2n A =  . . .. .   . . . .  αn1 αn2 . . . αnn

It is possible to generalize the tangent space to arbitrary quasi-projective varieties, but for this a more abstract definition is neccessary. First we will show, that there is a more abstract way to define the tangent space of an affine variety.

Definition 20.3 (Second Definition of Tangent space) Let X be an affine variety, with coor- dinate ring Γ(X). For p ∈ X let

mp = {f ∈ Γ(X) | f(p) = 0} be the corresponding maximal ideal. Then

2 ∨ Tp(X) = (mp/mp)

The compatibility to our previous definition comes from the following Claim 20.4 When using the definition of tangent space given in 20.1, we have a canonical isomorphism ∼ 2 ∨ Tp(X) = (mp/mp)

Proof. Suppose X ⊂ Cn.The map

B : Tp(X) × C[X1,...,Xn] → C

(v, f) 7→ (Dvf)p

By definition of the tangent space for every v ∈ Tp(X) the linear map B(v, ·) has kernel I(X). Thus by factoring we get a bilinear map B : Tp(X) × Γ(X) → C, and by restricting its domain 2 B : Tp(X) × mp → C. Now let v ∈ Tp(X) and f ∈ mp. Then f = f1g1 + ... + fkgk, for fi, gi ∈ mp, i = 1, . . . , k. Then by linearity of differentiation, Leibniz rule and as fi(p) = gi(p) = 0 for all i k X B(v, f) = (Dvf)p = (Dvfi)pgi(p) + fi(p)(Dvgi)p = 0. i=1 2 That means we can factor by mp which finally gives the bilinear:

2 B : Tp(X) × mp/mp → C.

29 This datum is equivalent to the linear map

2∨ L : Tp(X) → mp/mp

v 7→ (f 7→ (Dvf)p) which we will now prove to be an isomorphism. Write p = (ξ1, . . . , ξn) such that mp = (X1 − ξ1,..., Xn − ξn). Assume v ∈ Tp(X) such that for all f ∈ mp (Dvf)p = 0. Then in particular

vi = (Dv(Xi − ξi))p = 0,

2 so v = 0. This proves injectivity. Now assume ψ : mp/mp → C is linear. Any f ∈ I(X) fulfills f(p) = 0 and hence can be written as

n X f = (Xi − ξi)ai + (higher terms). i=1

As the image of f in Γ(X) is zero, ψ(f) = 0. At the same time (Xi −ξi)1≤i≤k span the C-vector 2 space mp/mp, and we have,

n n X X ψ(f) = ψ( (Xi − ξi)ai) = ψ(Xi − ξi)ai. i=1 i=1

Let vi = ψ(Xi − ξi) and n X v = vi∂i. i=0

Then by what we have shown it follows that v ∈ Tp(X) and that L(v) = ψ. This proves surjectivity of L. We now want to generalize this definition in a straightforward way to arbitrary quasi- projective varieties. One possibility would be to choose for each point an affine open neighbor- hood and apply the second definition. However, we will use a more invariant definition, which clearly does not depend on a choice. The main problem is how to find the appropriate ring and maximal ideal so that we can do the same as in Def. 20.3. The solution comes with the following

Definition 20.5 For any quasi-projective variety X and p ∈ X, we define the ring of germs of algebraic functions of X at p as

Op := {(U, f) | p ∈ U ⊂ X is open, f is an algebraic function on U.}/ ∼

Where the equivalence relation ∼ is defined as:

(U, f) ∼ (V, g) ⇔ ∃W ⊂ U ∩ V : W open and f|W = g|W

It is an easy exercise that together with the induced addition and multiplication, this is indeed a local ring where the maximal ideal consists of the classes of functions that vanish at p.

30 Definition 20.6 (Third and last definition of the Tangent space) Let X be any quasi-projective variety and p ∈ X. Let Op be the ring of germs of algebraic functions at p, mp its maximal ideal. Then the tangent space of X at p is

2∨ Tp(X) := mp/mp .

Again, we will show that this is compatible with our previous definitions: Claim 20.7 Let X is a quasi-projective variety, p ∈ X. Let Tp(X) be the Tangent space at p as in Def. 20.6. If V ⊂ X is an affine open subset and p ∈ V , there is a canonic isomorphism between Tp(V ) as in Def. 20.3 and Tp(X). Proof. We will use the following commutative algebra lemma which you can check for yourself:

Lemma 20.8 Let R be ring and m ⊂ R a maximal ideal. Any R/m vector space M is ∼ isomorphic as an R- to its localization: M = M ⊗R Rm

As restriction to an open neighborhood does not change the ring of functions, we can assume that X = V . Let A = Γ(V ) be the ring of functions of the affine variety V and mp the maximal ∼ ideal belonging to p. By exercise sheet 8, Ex. 2, we know that Op = Amp . Let mp the image of mp in OK . Now we consider the exact sequence of A-modules:

2 2 0 → mp → mp → mp/mp → 0.

Localization at mp gives:

2 2 0 → mp ⊗ Amp → m˜ p → mp/mp ⊗ Amp → 0

2 2 Now you should note that mp ⊗ Amp = m˜ p and use the lemma to get:

2 ∼ 2 mp/mp = m˜ p/m˜ p.

Taking the dual on both sides yields the isomorphism of the tangent spaces.

21 Tangent Space II 22 Blow-up

As a motivation for the blowup construction, consider the polar coordinates

2 ρ : R × [0, π]/ ∼ −→ R , ρ :(r, φ) 7→ (r cos(φ), r sin(φ)), where (r, 0) ∼ (−r, π). Note that the domain is a bit unusual, but on U = {r 6= 0}, ρ is isomorphic to the classical polar coordinates defined on R>0 × S1. Note the characteristic feature of ρ: It is an isomorphism on U = {r 6= 0} and contracts the circle r = 0 onto the zero-point. Polar coordinates are used to simplify calculations in calculus. We consider an analog construction in algebraic geometry, the blowup.

31 The blowup of A2 at 0 is defined by

2 2 1 Bl0A = {(x, y), [u, v] ∈ A × P | xv − yu = 0} together with the natural map 2 2 ρ : Bl0A −→ A 2 given by projection on the first factor. It is easy to see that ρ is an isomorphism on Bl0 A \ −1 −1 ∼ 1 {ρ (0, 0)}, while it contracts the central fiber ρ (0, 0) = P to the point (0, 0). In fact, when considering the blowup construction over the real numbers R, we obtain the 1 motivating example above (up to an isomorphism; use RP = S1/ ± 1 = [0, π]/ ∼). Then further discussion of the blowup, see Harris, Shafarevich, Hartshorne. Explicit com- putations are given in the solutions to sheet 9.

23 Dimension I

Topic is the equivalence of the definitions and several things about transcendence degree. Christoph recommended hungerford’s algebra for more details.

24 Dimension II 25 Sheaves

Let X be a topological space. A presheaf of abelian groups (or rings, etc.) on X is a contravari- ant functor F from the poset of open subsets of X to the category of abelian groups (rings, etc.). It is standard to denote F (U) by Γ(U, F ) (whose elements are called sections of F over U), and, for V ⊂ U, to denote the image of s ∈ F (U) under F (U) → F (V ) by s|V . Roughly speaking, a sheaf is a presheaf whose sections are determined locally, and compati- ble sections can be patched together; it establishes relations between local and global properties of a space. Formally, a sheaf on X is a presheaf F on X such that for every open subset U of X and every open covering {Ui} of U the following hold: (i) if s ∈ F (U) and s|Ui = 0 for all i then s = 0; (ii) if {si} is a family of sections si ∈ F (Ui) with si|Ui∩Uj = sj|Ui∩Uj for every i, j then there is a section s ∈ F (U) such that s|Ui = si for all i. (By (i) the section s must be unique.) Another way of putting this (giving a clue of how to generalize to sheaves with values in less concrete categories): we have an equalizer diagram Y Y F (U) → F (Ui) ⇒ F (Ui ∩ Uj)

were the first map is given by s 7→ {s|Ui } and the the other maps are {si} 7→ {si|Ui∩Uj } and

{si} 7→ {sj|Ui∩Uj }. (Sometimes, for instance in analysis books, a different definition of sheaves is used, involving the espace ´etal´e.) Let F be a presheaf on X. Notice that if x ∈ X is a point, the set of open neighborhoods U of x is directed under reverse inclusion; this gives a direct system F (U) (where the maps are the restrictions). The stalk of at x, denoted , is the direct limit lim (U) of this direct F Fx −→ F system. The construction (using the disjoint union, see exercise sheet 3 of the commutative algebra course) used to demonstrate the existence of the direct limit shows that we may think of Fx as having as elements equivalence classes [(s, U)] with U an open neighborhood of x and s ∈ F (U) (germs of sections of F at x); [(s, U)] and [(t, V )] are considered as equal if there is an open neighborhood W ⊂ U ∩ V of x with s|W = t|W . If U is an open neighborhood of x, the canonical map F (U) → Fx belonging to the direct limit is by taking the germ at x,

32 Q s 7→ [(s, U)]; these maps induce a map F (U) → x∈U Fx which is injective whenever F is a sheaf. Example 25.1 If X is a variety (topological space, C∞-manifold), then we have (in the obvious way) a sheaf of regular (continuous, C∞-) functions on X. The reason that this is a sheaf is that the notion of a regular (continuous, differentiable) function is defined locally. Example 25.2 The constant sheaf associated to an abelian group, the skyscraper sheaf. Example 25.3 Let X and Y be spaces, U an open subset of X. A sheaf F on X induces one on U, denoted F |U. If f : X → Y is a continuous map, the direct image sheaf f∗F of F is the sheaf on Y given by the composite of F and the functor from the category of open subsets of Y to the category of open subsets of X by taking the preimage under f. A morphism of (pre-)sheaves is simply a functorial morphism. Every morphism φ : F → G of presheaves on X gives naturally rise to notions such as its kernel, cokernel, image, and so on (see exercise sheet 11); also it induces (by the functoriality of lim, noticing that every morphism −→ of presheaves gives a morphism of direct systems) for every x ∈ X a map φx : Fx → Gx of stalks (which is explicitly given by φx([(s, U)]) = [(φU (s),U)]). A sequence F → G → H of presheaves on X is exact iff F (U) → G (U) → H (U) is exact for all open subsets U of X. However, the presheaf cokernel of a morphism of sheaves need not be a sheaf (but the presheaf kernel is one); this leads to sheafificiation (see exercise sheet 11). A sequence F → G → H of sheaves on X is exact iff it is exact at all stalks. Example 25.4 The sequence of sheaves of abelian groups on C exp 0 → 2πi → Ohol −−→Ohol,∗ → 0 Z C C is exact since it is exact at stalks (every nowhere vanishing holomorphic function is locally the exponential of some holomorphic function). The sequence of sections need, however, not be exact: for exp 0 → 2πi → Ohol( ∗) −−→Ohol,∗( ∗) → 0 Z C C C C hol ∗ hol,∗ ∗ cannot be exact as exp : O ( ) → O ( ) is not surjective (id ∗ is not in the image, as C C C C C there is no global logarithm on C∗).

26 Schemes I

Motivation for schemes: capturing nilpotents (varieties ignore nilpotents, e.g. V (x) = V (x2), the rings of algebraic functions are reduced); algebraic geometry over ground fields that are not algebraically closed, such as finite fields (the word on the street is that algebraic geometry over finite fields comes up in number theory, apparently this leads to some kind of unification of geometry and arithmetic). A is a pair (X, OX ) consisting of a space X and a sheaf of rings OX on X; OX is ] said to be the structure sheaf of X. A morphism of ringed spaces is a pair (f, f ):(X, OX ) → ] (Y, OY ) consisting of a continuous map f : X → Y and a morphism f : OY → f∗OX of sheaves on Y .A locally ringed space is a ringed space such that all stalks of the structure sheaf are local rings. A morphism of locally ringed spaces is a morphism of ringed spaces ] ] (f, f ):(X, OX ) → (Y, OY ) such that for every x ∈ X the canonical map fx : OY,f(x) → OX,x ] ] −1 explicitly given by fx([(s, U)]) = [(fU (s), f (U))] is a local homomorphism. cont cont Example 26.1 If X is a space, then (X, OX ) is a locally ringed space, since OX,x is a local ring for every x (the maximal ideal being the germs of functions vanishing at x). Similiarly for sm ∞ ∞ (X, OX ), X denoting a C -manifold. In fact a C -manifold of dimension n can be viewed as a locally R-ringed second-countable Hausdorff topological space which is locally isomorphic (as n sm a locally -ringed space) to ( , O n ). R R R

33 Let A be a ring, consider the topological space Spec A; its definition and some of its topological properties were mentioned in the exercises of the commutative algebra course. Most notably Spec A ≈ Spec Ared, so that the topology on Spec A is insensitive to nilpotents; this is captured by the structure sheaf. The definition of the structure sheaf of Spec A is described in Hartshorne. An affine scheme is a locally ringed space that is isomorphic to Spec A for some ring A.A scheme is a locally ringed space (X, OX ) which admits an open covering {Ui} such that every (Ui, OX |Ui) is an affine scheme. n Example 26.2 The affine scheme C is defined to be Spec C[x1, . . . , xn].

27 Schemes II

Here he just proved Prop. 2.2. and Prop. 2.3. in Hartshorne. The main point is that the properties established in Prop. 2.2. are nice to have (in the lecture the definition of the structure sheaf of Spec A was motivated by these properties), and that Prop. 2.3. comes as a great relief since it is much easier to think about the (less-structured) than the highly-structured category of affine schemes. (Here a comparision with the fairly involved definition of a C∞-manifold was drawn.)

34