SOME AMUSING PROBLEMS RELATED TO ALGEBRAIC GEOMETRY

RAVI VAKIL

Here are some problems to pique your interest; we will solve (or mostly solve) them during the semester. Some are deep, and others are just fun. You won’t actually need algebraic geometry to solve a fair number of these problems, but if you solve them (or make progress), you’ll secretly pick up a lot of algebro-geometrical insight.

1. Infinite Chomp. The game of Chomp is played as follows. Fix non-negative integers m and n. Cookies are placed in a rectangular array at the points (x, y) where 0 ≤ x ≤ m and 0 ≤ y ≤ n (x, y are integers). The cookie at (0,0) is poisoned. Two players alternate moving; a move involves picking a cookie, and eating it and every cookie above and to the right of it. The player who dies loses. (There’s a neat proof that if m and n aren’t both zero, the first player has a winning strategy, that doesn’t reveal what that strategy is.) Clearly the game ends in a finite number of terms.

Infinite Chomp is the same, except cookies are placed at (x, y) where x and y run through all non-negative integers. Prove that the game is guaranteed to end in a finite number of terms. Generalize this to where cookies are placed on (ℤ^{+})^{n} for any n.

(This is secretly related to the Hilbert Basis Theorem.)

2. Suppose f_{1}(x_{1}, …, x_{n}) = 0, …, f_{r}(x_{1}, …, x_{n}) = 0 is a system of r polynomial equations in n unknowns, with integral coefficients, and suppose this system has a finite number of complex solutions. Show that each solution is algebraic, i.e. if (x_{1}, …, x_{n}) is a solution, then x_{i} ∈ \overline{Q} for all i.

3. Pascal’s theorem. If a hexagon is inscribed in an irreducible conic, then the opposite sides meet in collinear points.

4. Pappas’ theorem. Let L_{1} and L_{2} be two lines. Let p_{1}, p_{2}, and p_{3} be distinct points of L_{1}, and let q_{1}, q_{2}, and q_{3} be distinct points of L_{2} (none lying on L_{1} ∩ L_{2}). Let L_{ij} be the line between P_{i} and Q_{j}. For each i, j, k with {i, j, k} = {1, 2, 3}, let R_{k} = L_{ij} ∩ L_{ji}. Then R_{1}, R_{2}, and R_{3} are collinear.

5. We’ll see that the integers ℤ should really be thought of as some weird curve.

Date: September 9, 1999.

6. Group law on an elliptic curve. We’ll define the group law on an elliptic curve, and see that it is associative. (We’ll first define elliptic curves, which secretly come up in the next three problems.)

7. Find all rational solutions to y^{2} = ax^{3} + bx^{2} where a, b ∈ ℚ.

8. Prove that if p(t) and q(t) are polynomials (with complex coefficients) such that p(t)^{2} = q(t)^{3} + 1, then p(t) and q(t) are both constant.

9. Poncelet’s theorem. Let C and D be two ellipses, with C contained in D. Pick a point p_{0} on C, and draw a tangent to D from p_{0}, which meets C again at some other point p_{1}. Repeat this process (picking the “other” tangent from p_{1}). Suppose after n repetitions, you return to your starting point, i.e. p_{n} = p_{0}. Prove that this is true no matter where you start.

10. If f(x) is a polynomial with integer coefficients, of degree d = 2g + 1 or 2g + 2, square-free mod p, then

|∑_{i=0}^{p−1} (\frac{f(i)}{p})| ≤ 2g\sqrt{p} + 1.

(The “fraction” denotes the symbol for quadratic residues.) If g > 1, the proof uses the Weil conjectures for curves. (In specific cases, an explicit formula can be given for the left-hand side.)

11. The degree of the discriminant b^{2} − 4ac of the quadratic ax^{2} + bx + c is 2. Show that the degree of the discriminant of a degree d polynomial is 2(d − 1).

12. Fix a positive integer d, and let σ_{1}, …, σ_{n} be n transpositions in S_{d} (i.e. they swap two elements of {1, …, d}). Suppose they generate S_{d}, and

σ_{1} … σ_{n} = e,

where e is the identity. Show that n ≥ 2(d − 1).