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Lecture Notes for the Algebraic Geometry Course Held by Rahul Pandharipande

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Lecture Notes for the Algebraic Geometry Course Held by Rahul Pandharipande Lecture Notes for the Algebraic Geometry 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 quadrics 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 polynomials 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 projective space 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 Commutative Algebra (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 C1- manifolds. (In algebraic geometry the local analysis of algebraic varieties is by commutative algebra, whereas in differential geometry the local analysis of C1-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) := fx 2 Cn j f(x) = 0 for all f 2 Xg 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 algebraic variety. By Hilbert's basis 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 ideal of S is the set of polynomials vanishing on S, I(S) := ff 2 C[x1; : : : ; xn] j f(x) = 0 for all x 2 Sg. 1 Both I and V are inclusion-reversing, and arep 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 Zariski topology 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 n U is a subvariety of V . Let us now consider the sets fUf g defined by f2C[x1;:::;xn] Uf := fx 2 V j f(x) 6= 0g. Remark 1.1 fUf g is a basis of the Zariski topology: let U ⊂ V be an open subset. f2C[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 polynomial f 2 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 2 C there exist an open subset Wp ⊂ U containing p and gp; hp 2 C[x1; : : : ; xn] such that 1.0 62 gp(Wp). 2. fj = 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 2 C there is an open neighborhood W of p in n and polynomial functions h and g such that fj = hp and p C p p Wp gp gp never vanishes on Wp. Inside Wp we can find a basic open set p 2 Urp ⊂ Wp, where n Urp = fρ 2 C j rp(ρ) 6= 0g: It follows that for all points p 2 we have a r 2 [x ; : : : ; x ] such that fj = hp and C p C 1 n Urp gp n 8q 2 C : gp(q) = 0 ! rp(q) = 0. p k With Hilbert's Nullstellensatz it follows that rp 2 (gp) and therefore rp = αpgp for some αp 2 C[x1; : : : ; xn]. On Urp we have that rp; gp; αp are never 0. On Urp we have h α h α fj = p p = p p Urp k gpαp rp 2 k n Define h^ = h α andr ^ = r . Then U = U k , and for every p 2 there is r with r (p) 6= 0 p p p p p rp rp C p p and fj = 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 9s1; : : : ; sm 2 C[x1; : : : ; xn]: siri = 1 i=1 hi We have fjU = 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 2 C[x1; : : : ; xn] We have to prove f 2 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 fj = i ;F j = 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 2 I(V ). We denote by Γ(V ) the ring of algebraic functions V ! C. Proposition 2.3 [x ; : : : ; x ] Γ(V ) ∼= C 1 n I(V ) Proof. By f 2 C[x1; : : : ; xn] 7! fjV : 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 ^ 2 Again, we have Ur \ Ur = Ur r . On Ur r we have = . Let hi = hiri; r^i = 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 n f(0; 0)g. fjU = 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 2 C[x; y], with (0; 0) 2 Ur, and there are h; g 2 C[x; y] such that x h 8(x; y) 2 U n f(0; 0)g : = 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 − y3jr(gx − hy) because (x2 − y3) is a radical ideal. x2 − y3 can't divide r since r(0; 0) 6= 0. Therefore x2 − y3jgx − hy Because g(0; 0) 6= 0, g has a nonzero constant term. Use the ring homomorphism 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.
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