Gathman

Question 1

Discuss solutions to y^2 = (x-1)(x-2)…(x-2n)

Answer 1

Surface of genus n-1
pg 4-5

Question 2

Discuss solutions to y^2 = f(x), f degree 2n

Answer 2

pg 5

Question 3

Discuss solutions to f(x,y) = 0, f degree d.

degree-genus formula?

Answer 3

Idea: deform polynomial to something easier to analyze –> product of d linear equations.

Union of d lines any 2 intersecting at a point… compactifying have S^2’s

d spheres, every 2 connect in a pair of points - d choose 2 connections. d-1 needed to create connected chain of spheres without loops. Every additional one adds a loop.

pg 6

Question 4

Why is enumerative geometry related to theoretical physics?

Answer 4

Questions like: does the surface contain a curve with specific property and how many?

string theory - elementary particles = one-dimensional. When particles move in time, sweep out a surface in space-time - this surface has natural complex structure coming from physical theory

pg 7-8

Question 5

Discuss the curve (t^3, t^4, t^5) in C^3.

Answer 5

Given parametrically. Can write in terms of polynomials - need 3 equations to cut out 1-dim object in C^3!

pg 8

Question 6

Define: affine space over k, Z(S), algebraic sets, I(X)

Answer 6

pg 11, 14

Question 7

Discuss definitions of Notherian rings. Prove equivalence

Answer 7

pg 12

Question 8

State and Prove Hilbert Basis Thm

Answer 8

pg 12

Question 9

Define Zariski topology and prove topology

Answer 9

pg 13

Question 10

State and prove Nullstellensatz over C

Answer 10

Thm. Maximal ideals of k[x1, … , xn] are exactly ideals of the form (x1-a1, … , xn-an)

Cor. 1-1 correspondence points <–> max ideals

pg 14-15

Question 11

Define radical ideal. Discuss/prove relationship between ideals and algebraic sets

Answer 11

1.2.9

Have bijection: algebraic sets <–> radical ideals

Think polynomials as ring of functions on A^n and how topological structure of A^n is precisely reflected in ring - can read off geometric info from ring of functions which is f.g.

pg 15-16

Question 12

Define: reducible, irreducible, affine variety, disconnected, connected

Answer 12

pg 16

Question 13

How is the fact that X is an affine variety reflected in the ideal of functions vanishing on it? Ring of functions on it?

Answer 13

Prime ideal, integral domain

pg 17

Question 14

Define: Notherian top space

prove irreducible decomp

Algebraic formulation?

Answer 14

DCC

every radical ideal is a finite intersection of prime ideals in a unique way

pg 17-18

Question 15

Define: dimension of Noetherian top space

Answer 15

longest chain

pg 18

Question 16

Prove in an irreducible top space, every non-empty open subset is dense

Answer 16

pg 19

Question 17

Exercise 1.4.1

Question 18

Exercise 1.4.2

Question 19

Exercise 1.4.3

Question 20

Exercise 1.4.4

Question 21

Exercise 1.4.5

Question 22

Exercise 1.4.6

Question 23

Exercise 1.4.8

Question 24

Exercise 1.4.9