Homework Assignments for Math 697B - Fall 2006

• Assignment 1   Due Thursday, September 14.
  • Section I.1 page 7:   F, H
  • Section I.2 page 12:   C, G (generalize using Eisenstein criterion), I
• Assignment 2   Due Thursday, September 21.
  • Section I.2 page 12:   B
  • Section I.3 page 18:   E, G, F. In F you need not determine the genus of the complete intersection curve C; show instead that x_2/x_0 is a meromorphic function on C, which extends to a well defined map from C to the projective line,   exactly four fibers of g consist of a single point, and all other fibers consist of two distinct points. It follows that C has genus 1, by the Riemann-Hurwitz formula, to be proven later in the course (Theorem II.4.16).
    Hint: You may want to guide your calculations by the fact (prove it if you use it!), that the quadric   x_0x_3=2x_1x_2   is the image of the cartesian product of two copies of the projective line, via the bijective map sending ([a : b],[t : u]) to [at : au/2 : bt : bu]. Under this identification, the map g is the restriction to C of the first projection.
• Assignment 3   Due Thursday, September 28.
  • Homework Assignment 3
• Assignment 4   Due Thursday, October 5.
  • Section II.3 page 43:   F, G, H, K
  • Section II.4 page 53:   C
• Assignment 5   Due Thursday, October 19.
  • Section II.4 page 53:   D, G, I, J
  • Section III.1 page 65:   C, F, H, J
• Assignment 6   Due Thursday, November 2.
  • Section IV.1 page 111:   D, E, G (but see the extra problem below), I
    Hint for problem E: You will need first to compare one of   a_1:=P(u,v,1)du/(df/dv)   or   a_2 := P(u,v,1)dv/(df/du) to a one form on another affine patch (say y=1). Set g(t,w)=F(t,1,w),   b_1:=P(t,1,w)dt/(dg/dw),   b_2:=P(t,1,w)dw/(dg/dt). Of the four possible equalities    a_i  =  +- b_j,   the one involving the least amount of computations is    a_2 = -b_2.
    Notes for problem E:
    1) (Interpretation) Show that you found a (d-1)(d-2)/2 dimensional vector space of holomorphic 1-forms. We will later prove, that the dimension of the vector space of global holomorphic one-forms on a compact Riemann-surface is always equal to its genus. In particular, you have found all the holomorphic one-forms for a smooth projective plane curve (the verification is postponed to problem VI.3.I page 193).
    2) (Generalizations) This problem can be solved more invariantly, using residues of meromorphic differential forms (of a 2-form over P^2 with a simple pole along the curve, in our case). This generalization plays an important role in Hodge theory and generalizes to higher dimentional complete intersections in projective spaces and in more general (toric) varieties.
  • Section IV.2 page 117:   H, I (but see the extra problem below)
  • A problem including a combination of both problem IV.1.G page 111 and problem IV.2.I page 117 and a little more (which in Miranda's book is postponed to problem VI.3.H page 193).
• Assignment 7   Due Thursday, November 9.
  • Section IV.3 page 124 to 126:   Read the summary on Homotopy and Homology.
  • Section IV.3 page 127:   E (Note that F and H were proven in class in the course of proving Abel's Theorem for the torus).
  • Section V.1 page 137:   A, E [Show also that if p is a ramification point, then
    (2g+2)p is linearly equivalent to the ramification divisor, and
    (2g-2)p is a canonical divisor. For the latter, use part 1 of the extra problem from assignment 6],
    F, G, H, I, J
  • Section V.2 page 145:   B, C, F
• Assignment 8   Due Thursday, November 16.
  • Extra problem for section V.2: The genus formula of a complete intersection curve.
  • Section V.3 page 152:   C, F, H
• Assignment 9   Due Thursday, November 30.
  • Section V.4 page 166:   C, F (done in class).
  • Section V.4 page 166:   A reformulation of problem J.
  • Section V.4 page 166:   K
  • Extra problems for section V.4: Projective normality of rational and elliptic normal curves.
  • Section VI.1 page 178:   B, F, I
• Assignment 10   Due Thursday, December 7.
  • Section VI.2 page 185:   E, F, G, H (done in class for any Riemann surface not isomorphic to the Riemann sphere), J
  • Section VI.3 page 193:   C, G, H (a solution was presented by Penny in the extra problem for homework 6 without using Riemann-Roch), I, J
• Assignment 11   Due Thursday, December 14.
  • Section VII.1 page 202:   B, C, (one problem: D, E, F)
  • Section VII.2 page 215:   A, F
  • Let X be a compact hyperelliptic Riemann surface, i.e., such that there exists a degree 2 holomorphic map F:X->P^1. Show that X is the compactification of the affine plane curve given by y^2=h(x), where h(x) is a polynomial with distinct roots. Compare with the compactification in Lemma III.1.7 page 59. Hint: Use Proposition VI.1.21 page 176. Note: The statement is proven in Proposition III.4.11 page 92 using another method.
  • Extra problems: to be included