Arithmetic and Geometry of del Pezzo surfaces of degree 1

Weak Approximation on del Pezzo surfaces of degree 1 (arxiv copy)

Abstract: We study the arithmetic of del Pezzo surfaces of degree 1 of the form w^2 = z^3 + Ax^6 + By^6 in the weighted projective space P_k(1,1,2,3), where k is a perfect field of characteristic not 2 or 3 and A,B are in k^*. We exhibit an infinite family of (minimal) counterexamples to weak approximation amongst these surfaces, via a Brauer-Manin obstruction.

Magma Scripts with all the relevant computations

Cox rings of degree one del Pezzo surfaces (with D. Testa and M. Velasco) (arxiv copy)

Abstract: Let X be a del Pezzo surface of degree one over an algebraically closed field (of any characteristic), and let Cox(X) be its total coordinate ring. We prove the missing case of a conjecture of Batyrev and Popov, which states that Cox(X) is a quadratic algebra. We use a complex of vector spaces whose homology determines part of the structure of the minimal free Pic(X)-graded resolution of Cox(X) over a polynomial ring. We show that sufficiently many Betti numbers of this minimal free resolution vanish to establish the conjecture.

Arithmetic E_8 lattices with maximal Galois action (with D. Zywina) (arxiv copy)

Abstract: We construct explicit examples of E_8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E_8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E_8 and have maximal Galois action.

Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E_8. We construct examples of such surfaces for which the action of Galois on the geometric Picard group is maximal.

Older Material

These are some Papers and Notes I wrote when I was an undergraduate. Only one of them contains original results, though several of the expositions have proofs that I came up with.

My Undergraduate Thesis (pdf)

My thesis, titled "Singular and Supersingular Moduli," is about absolute norms of singular moduli (j-invariants of elliptic curves with complex multiplication). These numbers are huge, yet their prime factors are tiny. My thesis explains in a `down to earth way' some results of Gross and Zagier related to this phenomenon. The results included are not new, but I had to reconstruct many proofs myself as a non-specialist. I'm kinda proud of it.

Primes of the form x^2 + ny^2 (pdf)

This paper was written for Math 251r: Arithmetic Theory of Quadratic Forms. It starts off with a bit of background in Class Field Theory to prove there is an algorithm to determine which primes are of the form x^2 + ny^2. Then there's a discussion of how to use lattices that admit complex multiplication in order to implement the algorithm.

Witt vectors (pdf)

This paper was written for Math 250b: Higher Algebra II. This was in fact a take-home final. We were given 16 questions on Witt vectors and we had to string them together into a coherent paper. We were allowed to consult as many sources as we wanted. This actually made the exam a bit more difficult ;) Anyway, the first part of the paper contains a proof of the existence of Witt rings and the second part exhibits a construction due to Witt.

The Kronecker-Weber Theorem (dvi, pdf,ps)

This paper was written for Math 250a: Higher Algebra. It contains an elementary proof of the Kronecker-Weber Theorem via Higher Ramification Groups of Number Fields. It also has an exposition of the necessary Algebraic Number Theory.

Dirichlet's Theorem on Arithmetic Progressions (dvi, pdf,ps)

I wrote this paper for a Tutorial during the Summer of 2001.

Location of Incenters and Fermat Points in Variable Triangles (dvi, pdf,ps)

This is the only paper on this page with original results. It was published in the April 2001 issue of Mathematics Magazine (it's also available here). The paper grew out of an investigation I did in High-School for the International Baccalaureate program. I investigated the loci of incenters of triangles with fixed Euler line. I also proved that the Fermat Point of a triangle is always inside the orthocentroidal disc.

Describing the Real Numbers (dvi, pdf,ps)

This is a cute little exposition I wrote up when I was a Course Assistant for Math 25a (Honors Multivariable Calculus and Linear Algebra I). I also used it with students in Math 112 (Real Analysis). The file contains a description of the real numbers: field and order axioms, nested intervals property (aka completeness) and Archimedian property. The file ends with a proof that any two objects that satisfy these axioms/properties are isomorphic.

Introduction to Graph Theory (dvi, pdf,ps)

These are the notes of a "Minicourse" in Graph Theory that I taught at PROMYS in the Summer of 2001. The audience consisted of very talented High-School students. The goal of the course was to prove the Five-Color Theorem (well...I had to assume the Jordan Curve Theorem, but hey, most High-School students are willing to accept this hard theorem as "obvious"). I can't get the figures here. Will work on this some day to fix this problem. Sorry.