Linear equivalence of divisors:
Problem 1. (5 points) Prove that the Picard group of P^2 is isomorphic to Z. Problem 2. (Addition of points on elliptic curve.) Let E be a plane curve given by homogeneous equation y^2 * z = x^3 + a x * z^2 + b * z^3. where a and b are numbers. a.) (2 points) Sketch the picture of the affine part of E where z <> 0. Sketch (symbolically) the picture of E in P^2. b.) Note that E contains a point (0,1,0). Let us call this point O. c.) (2 points) Prove that there is a line l_0 on P^2 such that l_0 is tangent to E "of order 2" at O, i.e., E geometrically "looks like" the graph of y = x^3 around (0,0). Such a point is called inflection point. d.) (5 points) Let p, q be two points on E given in coordinates as (x,y,1) and (x,-y,1). Prove that q ~ -p + 2 O, where "~" is "linearly equivalent". In particular, if we say (by definition), that the point O is zero, we have q = -p in the Picard group Pic E. Hint: consider the vertical line. e.) (e + f = 6 points) Let l be a line intersecting E at 3 points, a,b and c. Prove that a + b + c ~ 3 O f.) In particular, if we let "O = 0", we get a + b = -c in Pic E. Combined with d.), this gives a purely geometric recipe for finding a + b in Pic E. (Draw a picture.) This gives a homomorphism of -groups- u: E -> Pic E. g.) (6 points) if p,q are two points on E, prove that it is impossible that p ~ q. It follows that u is injective. (In reality, one has Pic E = E x Z).
Singularities of curves and surfaces:
Problem 3. (6 points) Resolve the singularity of plane (affine) curve y^2 = x^4. Draw the picture of the resolution, including the exceptional curves. Problem 4. (8 points) (Icosahedral singularity).
Resolve the singularity of a surface (in A^3) x^2 + y^3 + z^5 = 0. Draw the corresponding (Dynkin) diagram. Problem 5. Study some chapters of the Felix Klein's book "Lectures on the Icosahedron" (or T. Springer's book "Invariant Theory") and check that a.) (6 points) There is a finite group G of order 120 acting linearly on C^2. b.) (7 points) The factor C^2/G is x^2 + y^3 + z^5. c.)^** (13 points) In reality, G is isomorphic SL_2(F_5) - linear group over the field of 5 elements.
Differential forms and adjunction.
Problem 6 (a)(7 points) Find the regular differential forms on a curve from problem 2. (b)(5 points) Find the canonical class of P^1 x P^1 by computing singularities of some rational differential form. Problem 7. Find the genus of a plane quartic curve (a)(5 points) By adjunction formula; (b)(5 points) By the Riemann -- Hurwitz formula. Problem 8.^*(10 points) Prove that intersection of 2 quadrics in P^3 is a curve of genus 1. (Hint: use adjunction formula twice).