Math 4800 – Spring 2015
Homework Set 2
1. Compute the points of intersection (in P2C ), and the corresponding multiplicities of intersection,
of the curves
C : x − y 2 = 0 and D : x3 − x2 + y 2 = 0.
2. Use the Resultant to show that ip (C, D) = 2, where
p = (0, 0),
C : y = x2
and D : y = 2x2 .
3. Give a second solution to the previous exercise, using instead Bezout’s Theorem and symmetries
in place of the Resultant to show that ip (C, D) = 2.
4. Show that ip (C, D) = 2 one more time using this time a parametrization of the conic C.
5. Compute the Hessian of the cubic C of equation y 2 = x3 − 1 and find all the inflection points
of C.
6. Prove that linear equivalence in the divisor group Div(C) of a (smooth projective) curve C is
an equivalence relation.
7. Show that any two points p, q ∈ P1 are linearly equivalent as divisors. Generalize this (or apply
this) to conclude that a divisor D on P1 is a principal divisor if and only if deg D = 0.
8. Argue that two distinct points p, q ∈ C on a (smooth projective) cubic curve are not linearly
equivalent. Show, on the other hand, that there are distinct points p, q ∈ C such that 2p ∼ 2q.
9. Prove that if ω and ω 0 are two meromorphic 1-forms on a (smooth projective) curve C then
div(ω) ∼ div(ω 0 ) (i.e., the canonical divisor of C is uniquely defined up to linear equivalence).
10. Let γ be the loop in C supported on the unit circle |z| = 1, given by the parameterization
z = eiθ , 0 ≤ θ ≤ 2π. Prove that
(
Z
2πi if n = −1
n
z dz =
0
otherwise
γ
∗∗∗
Possible topics for independent projects:
• Weierstrass P-function.
• Elliptic integrals.
• Proof of the Riemann–Roch theorem.
• Geometric proof of the Lüroth theorem.
• Elliptic curves and cryptography.
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