Algebraic Geometry, autumn term 2018 Christian B¨ohning Mathematics Institute University of Warwick ii Contents 1 Affine and projective algebraic sets 1 2 Algebraic varieties and regular maps 9 3 Hilbert's Nullstellensatz, primary decomposition 17 4 Segre embeddings, Veronese maps and products 27 5 Grassmannians, flag manifolds, Schubert varieties 35 6 Images of projective varieties under morphisms 45 7 Finite morphisms, Noether normalization 51 8 Dimension theory 59 9 Lines on surfaces, degree 69 10 Regular and singular points, tangent space 81 11 Cubic surfaces and their lines 91 12 Local parameters, power series methods, divisors 101 iii iv CONTENTS Chapter 1 Affine and projective algebraic sets; rational normal curves, finite point sets Below, k is an algebraically closed field of arbitrary characteristic, e.g. C, Q¯ ¯ n n or also Fp, and A is the n-dimensional affine space over k, i.e. k as a set (though we will want to make use of its vector space structure occasionally as well). Definition 1.1. For a k-vector space, P(V ) denotes the set of 1-dimensional subspaces of V , i.e. the projective space associated to V . For the vector space kn+1 with componentwise addition and scalar multiplication, we also write Pn = P(kn+1). Equivalently, Pn is the quotient of kn+1 − f0g by the equivalence relation: 0 0 0 0 n+1 (X0;:::;Xn) ∼ (X0;:::;Xn); (X0;:::;Xn); (X0;:::;Xn) 2 k − f0g ∗ 0 0 if there is a λ 2 k with λ(X0;:::;Xn) = (X0;:::;Xn). We denote the equivalence class [(X0;:::;Xn)] by (X0 : ··· : Xn) in that case and call the n Xi's homogeneous coordinates of the point in P . n Define a subset Ui ⊂ P by n Ui := f(X0 : ··· : Xn) 2 P j Xi 6= 0g : n Below we will often identify Ui with A via the bijection (X0 : ··· : Xn) 7! (i) (i) (xj )0≤j≤n = (Xj=Xi) (so xi = 1). 1 2 CHAPTER 1. AFFINE AND PROJECTIVE ALGEBRAIC SETS Definition 1.2. 1. An affine algebraic set X ⊂ An is the set of zeroes of a family (fα)α2A, of polynomials fα 2 k[x1; : : : ; xn]. Since k[x1; : : : ; xn] is Noetherian, we can assume jAj < 1 without loss of generality. 2. A projective algebraic set Y ⊂ Pn is the set of zeroes of a family of polynomials (Fα)α2A with Fα 2 k[X0;:::;Xn]. We have to say a few words what it means to be a zero in (2) above, n i.e. what is meant by F (p) = 0 for a p = (P0 : ··· : Pn) 2 P . This is so because the homogeneous coordinates Pi of p are not unique, and p being a zero must be independent of the representative coordinate tuple. We do this by defining F (p) = 0 : () F (P0;:::;Pn) = 0 n+1 for all (P0;:::;Pn) 2 k − f0g with [(P0;:::;Pn)] = p: This leads to the conclusion that we can assume, without loss of gen- erality, that the Fα in (2) of Definition 1.2 are homogeneous, which means the following: S = k[X0;:::;Xn] is a graded ring, which means there is a decomposition into k-vector subspaces M S = Sm m≥0 α0 αn where Sm := hX0 ·:::·Xn ik, α0 +: : : αm = m, such that Sm1 ·Sm2 ⊂ Sm1+m2 . Polynomials in Sm are called homogeneous of degree m. Now, since k is infinite, if F 2 k[X0;:::;Xn] vanishes in p as above, then all homogeneous components Fm of F with respect to the preceding direct sum decomposition vanish in p. n n Remark 1.3. If Y ⊂ P is a projective algebraic set, then Yi = Y \Ui ⊂ A is an affine algebraic set. To see this, consider for simplicity Y0; the argument in the other cases being the same. If Y is defined by homogeneous polynomials Fα of degree dα, then Y0 is the set of zeroes of polynomials fα(x1; : : : ; xn), xi = Xi=X0, where dα fα(x1; : : : ; xn) := Fα(X0;:::;Xn)=X0 = Fα(1; x1; : : : ; xn): n n Remark 1.4. Every affine algebraic subset Xi ⊂ A ' Ui ⊂ P is the intersec- n tion of Ui with a projective algebraic subset X ⊂ P . Again we show that for 3 U0 only since the other cases are only notationally different. If X0 is defined by fα(x1; : : : ; xn) of degree dα (of course not necessarily homogeneous now), we can define a suitable X by dα X1 Xn Fα(X0;:::;Xn) := X0 fα ;:::; : X0 X0 We can summarize the preceding two remarks by saying that X ⊂ Pn is n a projective algebraic subset if and only if each X \Ui ⊂ Ui ' A is an affine algebraic set. Example 1.5. 1. If W ⊂ kn+1 is an (m+1)-dimensional sub-vector space, then P(W ) ⊂ Pn is a projective algebraic set, called an m-dimensional projective linear subspace (for m = 1: line, m = 2: plane, m = n − 1: hyperplane). 2. Of course zeroes of a single homogeneous polynomial F 2 k[X0;:::;Xn] (of degree d, say) are a projective algebraic set; this is called a hy- persurface. We can assume F without multiple factors (note that k[X0;:::;Xn] is factorial). Then we call d the degree of the hyper- surface. 3. For a more interesting example, let C be the image of the map 1 3 ν : P ! P 3 2 2 3 (X0 : X1) 7! (X0 : X0 X1 : X0X1 : X1 ) := (Z0 : Z1 : Z2 : Z3): Then C is contained in the three quadrics Q0;Q1;Q2 defined as the zero sets of 2 F0(Z) = Z0Z2 − Z1 F1(Z) = Z0Z3 − Z1Z2 2 F2(Z) = Z1Z3 − Z2 and those define C, i.e. the zero set of them is exactly C (to see this, 3 note that if p 2 P , p = (P0 : P1 : P2 : P3) lies on Q0 \ Q1 \ Q2, then P0 6= 0 or P3 6= 0; in the former case, P = ν((P0 : P1)), in the latter case, P = ν((P2 : P3)). 4 CHAPTER 1. AFFINE AND PROJECTIVE ALGEBRAIC SETS We continue the study of (3) in the above examples a little: although we haven't introduced any notion of \dimension" yet into our geometric study of algebraic sets, it is intuitively plausible that C should be a one-dimensional thing, a curve. It is called a twisted cubic curve. It is remarkable that two of the above quadratic equations do not define C, more generally: 3 Proposition 1.6. For λ = (λ0; λ1; λ2) 2 k − f0g, write Fλ = λ0F0 + λ1F1 + λ2F2; the Fi being as in Example 1.5, (3). Denote the projective algebraic set that 2 Fλ defines by Qλ. Then for [µ] 6= [ν] 2 P , we have that Qµ \ Qν is equal to the union of C and a line Lµν intersecting C in two points. Proof. C is defined by the 2 × 2 minors of Z0 Z1 Z2 Z1 Z2 Z3 and Qµ is the determinant of 0 1 Z0 Z1 Z2 @Z1 Z2 Z3A 0 0 0 µ0 µ1 µ2 0 0 0 where the tuple (µ0; µ1; µ2) agrees with µ after a signed permutation. So the locus outside of C where Fµ and Fν vanish, is the rank ≤ 2 locus of 0 1 Z0 Z1 Z2 BZ1 Z2 Z3C B 0 0 0 C @µ0 µ1 µ2A 0 0 0 ν0 ν1 ν2 (where in addition the first two rows are independent). For [µ] 6= [ν], this locus is the same as the one defined by 0 1 0 1 Z0 Z1 Z2 Z1 Z2 Z3 0 0 0 0 0 0 det @µ0 µ1 µ2A = det @µ0 µ1 µ2A = 0; 0 0 0 0 0 0 ν0 ν1 ν2 ν0 ν1 ν2 i.e. Qµ \ Qν = C [ Lµν where Lµν is the line defined by the last two deter- minants. The intersection of Lµν with C is then given by Qλ and the two linear equations above where hQλ;Qµ;Qνi = hF0;F1;F2i. 5 Proposition 1.7. There exists a homogeneous quadratic polynomial Q(Z0;:::;Z3) and a homogeneous cubic polynomial P (Z0;:::;Z3) whose common zeroes are precisely C. Proof. One can take 0 1 Z0 Z1 Z2 Z0 Z1 Q(Z) = det ;P (z) = det @Z1 Z2 Z3A : Z1 Z2 Z2 Z3 Z0 Namely, if the vector (Z0;Z1;Z2) and (Z1;Z2;Z3) are linearly dependent, then these determinants vanish, and the converse holds as well: if the first two rows of the matrix whose determinant defines P (z) are independent, and the determinant vanishes, then the last row of the matrix must be a linear combination of the first two rows. But then (Z2;Z3) is dependent on (Z0;Z1) and (Z1;Z2), whence the rank of the submatrix consisting of the first two rows would be 1 (taking into account Q(z) = 0), contradiction. Thus we arrive at the curious fact that C, as a set, can be defined by two polynomials, but if we look at the ideal I(C) ⊂ k[X0;:::;X3] of all polynomials vanishing on C, this cannot be generated by 2 elements (since dim I(C)2 ≥ 3, but dim I(C)1 = 0: clearly, C does not lie in a hyper- 3 2 2 3 plane since X0 ;X0 X1;X0X1 ;X1 are independent). One says that C is a set- theoretic complete intersection, but not a complete intersection in P3. There are many open problems connected with these notions; e.g., one knows that the union of two planes intersecting only in 0 is not a set-theoretic complete intersection in A4, but one does not know if every curve in P3 is a set-theoretic complete intersection.