Part II | Algebraic Geometry | Year 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2021 3 Paper 1, Section II 25I Algebraic Geometry Let k be an algebraically closed field and let V An be a non-empty affine variety. ⊂ k Show that V is a finite union of irreducible subvarieties. n Let V1 and V2 be subvarieties of Ak given by the vanishing loci of ideals I1 and I2 respectively. Prove the following assertions. (i) The variety V V is equal to the vanishing locus of the ideal I + I . 1 ∩ 2 1 2 (ii) The variety V V is equal to the vanishing locus of the ideal I I . 1 ∪ 2 1 ∩ 2 Decompose the vanishing locus 2 2 2 2 3 V(X + Y 1,X Z 1) A . − − − ⊂ C into irreducible components. 3 Let V Ak be the union of the three coordinate axes. Let W be the union of three ⊂ 2 distinct lines through the point (0, 0) in Ak. Prove that W is not isomorphic to V . Paper 2, Section II 25I Algebraic Geometry Let k be an algebraically closed field and n > 1. Exhibit GL(n, k) as an open subset n2 of affine space Ak . Deduce that GL(n, k) is smooth. Prove that it is also irreducible. Prove that GL(n, k) is isomorphic to a closed subvariety in an affine space. Show that the matrix multiplication map GL(n, k) GL(n, k) GL(n, k) × → that sends a pair of matrices to their product is a morphism. Prove that any morphism from An to A1 r 0 is constant. k k { } n 1 Prove that for n > 2 any morphism from Pk to Pk is constant. Part II, 2021 List of Questions [TURN OVER] 2021 4 Paper 3, Section II 24I Algebraic Geometry In this question, all varieties are over an algebraically closed field k of characteristic zero. What does it mean for a projective variety to be smooth? Give an example of a smooth affine variety X An whose projective closure X Pn is not smooth. ⊂ k ⊂ k 4 What is the genus of a smooth projective curve? Let X Pk be the hypersurface 3 3 3 3 3 ⊂ V (X0 + X1 + X2 + X3 + X4 ). Prove that X contains a smooth curve of genus 1. Let C P2 be an irreducible curve of degree 2. Prove that C is isomorphic to P1. ⊂ k k 2 We define a generalized conic in Pk to be the vanishing locus of a non-zero homogeneous quadratic polynomial in 3 variables. Show that there is a bijection between 2 5 the set of generalized conics in Pk and the projective space Pk, which maps the conic V (f) to the point whose coordinates are the coefficients of f. 5 (i) Let R◦ P be the subset of conics that consist of unions of two distinct ⊂ k lines. Prove that R◦ is not Zariski closed, and calculate its dimension. (ii) Let I be the homogeneous ideal of polynomials vanishing on R◦. Determine generators for the ideal I. Paper 4, Section II 24I Algebraic Geometry Let C be a smooth irreducible projective algebraic curve over an algebraically closed field. Let D be an effective divisor on C. Prove that the vector space L(D) of rational functions with poles bounded by D is finite dimensional. Let D and E be linearly equivalent divisors on C. Exhibit an isomorphism between the vector spaces L(D) and L(E). What is a canonical divisor on C? State the Riemann–Roch theorem and use it to calculate the degree of a canonical divisor in terms of the genus of C. Prove that the canonical divisor on a smooth cubic plane curve is linearly equivalent to the zero divisor. Part II, 2021 List of Questions 2020 3 Paper 1, Section II 25F Algebraic Geometry Let k be an algebraically closed field of characteristic zero. Prove that an affine variety V An is irreducible if and only if the associated ideal I(V ) of polynomials that ⊂ k vanish on V is prime. Prove that the variety V(y2 x3) A2 is irreducible. − ⊂ k State what it means for an affine variety over k to be smooth and determine whether or not V(y2 x3) is smooth. − Paper 2, Section II 24F Algebraic Geometry Let k be an algebraically closed field of characteristic not equal to 2 and let V P3 ⊂ k be a nonsingular quadric surface. 2 (a) Prove that V is birational to Pk. (b) Prove that there exists a pair of disjoint lines on V . (c) Prove that the affine variety W = V(xyz 1) A3 does not contain any lines. − ⊂ k Part II, 2020 List of Questions [TURN OVER] 2020 4 Paper 3, Section II 24F Algebraic Geometry (i) Suppose f(x, y) = 0 is an affine equation whose projective completion is a smooth projective curve. Give a basis for the vector space of holomorphic differential forms on this curve. [You are not required to prove your assertion.] Let C P2 be the plane curve given by the vanishing of the polynomial ⊂ X4 X4 X4 = 0 0 − 1 − 2 over the complex numbers. (ii) Prove that C is nonsingular. (iii) Let ` be a line in P2 and define D to be the divisor ` C. Prove that D is a ∩ canonical divisor on C. (iv) Calculate the minimum degree d such that there exists a non-constant map 1 C P → of degree d. [You may use any results from the lectures provided that they are stated clearly.] Paper 4, Section II 24F Algebraic Geometry Let P0,...,Pn be a basis for the homogeneous polynomials of degree n in variables 1 n Z0 and Z1. Then the image of the map P P given by → [Z ,Z ] [P (Z ,Z ),...,P (Z ,Z )] 0 1 7→ 0 0 1 n 0 1 is called a rational normal curve. n Let p1, . , pn+3 be a collection of points in general linear position in P . Prove that there exists a unique rational normal curve in Pn passing through these points. Choose a basis of homogeneous polynomials of degree 3 as above, and give generators for the homogeneous ideal of the corresponding rational normal curve. Part II, 2020 List of Questions 2019 3 Paper 4, Section II 24F Algebraic Geometry (a) Let X P2 be a smooth projective plane curve, defined by a homogeneous ⊆ polynomial F (x,y,z) of degree d over the complex numbers C. (i) Define the divisor [X H], where H is a hyperplane in P2 not contained in ∩ X, and prove that it has degree d. (ii) Give (without proof) an expression for the degree of in terms of d. KX (iii) Show that X does not have genus 2. (b) Let X be a smooth projective curve of genus g over the complex numbers C. For p X let ∈ G(p)= n N there is no f k(X) with vp(f)= n, and vq(f) 0 for all q = p . { ∈ | ∈ 6 } (i) Define ℓ(D), for a divisor D. (ii) Show that for all p X, ∈ ℓ((n 1)p) for n G(p) ℓ(np)= − ∈ ℓ((n 1)p) + 1 otherwise. − (iii) Show that G(p) has exactly g elements. [Hint: What happens for large n?] (iv) Now suppose that X has genus 2. Show that G(p)= 1, 2 or G(p)= 1, 3 . { } { } [In this question N denotes the set of positive integers.] Part II, 2019 List of Questions [TURN OVER 2019 4 Paper 3, Section II 24F Algebraic Geometry Let W A2 be the curve defined by the equation y3 = x4 + 1 over the complex ⊆ numbers C, and let X P2 be its closure. ⊆ (a) Show X is smooth. (b) Determine the ramification points of the map X P1 defined by → (x : y : z) (x : z). → Using this, determine the Euler characteristic and genus of X, stating clearly any theorems that you are using. (c) Let ω = dx . Compute ν (ω) for all p X, and determine a basis for y2 ∈ KX p ∈ ( ). L KX Paper 2, Section II 24F Algebraic Geometry (a) Let A be a commutative algebra over a field k, and p : A k a k-linear → homomorphism. Define Der(A, p), the derivations of A centered in p, and define the tangent space TpA in terms of this. Show directly from your definition that if f A is not a zero divisor and p(f) = 0, ∈ then the natural map T A[ 1 ] T A is an isomorphism. p f → p (b) Suppose k is an algebraically closed field and λ k for 1 i r. Let i ∈ 6 6 2 2 X = (x,y) A x = 0,y = 0,y = (x λ1) (x λr) . { ∈ | − · · · − } Find a surjective map X A1. Justify your answer. → Part II, 2019 List of Questions 2019 5 Paper 1, Section II 25F Algebraic Geometry (a) Let k be an algebraically closed field of characteristic 0. Consider the algebraic variety V A3 defined over k by the polynomials ⊂ xy, y2 z3 + xz, and x(x + y + 2z + 1). − Determine (i) the irreducible components of V , (ii) the tangent space at each point of V , (iii) for each irreducible component, the smooth points of that component, and (iv) the dimensions of the irreducible components. n (b) Let L K be a finite extension of fields, and dimK L = n. Identify L with A ⊇ over K and show that U = α L K[α]= L { ∈ | } is the complement in An of the vanishing set of some polynomial.