Arithmetic and Geometry of del Pezzo surfaces of degree 1

Density of rational points on isotrivial rational elliptic surfaces   Submitted.   (arxiv copy)

Abstract: For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We also prove that these surfaces satisfy a variant of weak-weak approximation. Our results are conditional on the finiteness of Tate-Shafarevich groups for elliptic curves over the field of rational numbers.


Weak Approximation on del Pezzo surfaces of degree 1   Adv. Math. 219 (2008), 2123-2145.   (arxiv copy)

Abstract: We study 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^*. Over a number field, 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


Arithmetic E_8 lattices with maximal Galois action (with D. Zywina)   LMS J. Comput. Math. 12 (2009) 144-165.   (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.


Cox rings of degree one del Pezzo surfaces (with D. Testa and M. Velasco)   Algebra and Number Theory 3 (2009) 729-761. (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.

Algebraic Geometry

Big rational surfaces (with D. Testa and M. Velasco)   Submitted.   (arxiv copy)

Abstract: We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of the plane at distinct points lying on a (possibly reducible) cubic.

Laurent polynomials and Eulerian numbers (with D. Erman and G. Smith)   Submitted.   (arxiv copy)

Abstract: Duistermaat and van der Kallen show that there is no nontrivial complex Laurent polynomial all of whose powers have a zero constant term. Inspired by this, Sturmfels posed two questions: Do the constant terms of a generic Laurent polynomial form a regular sequence? If so, then what is the degree of the associated zero-dimensional ideal? In this note, we prove that the Eulerian numbers provide the answer to the second question. The proof involves reinterpreting the problem in terms of toric geometry.

Older Material

Location of Incenters and Fermat Points in Variable Triangles   Math. Mag. 74 (2001) 123-129.   (pdf)

This note 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.

Expository Papers

These are some Papers and Notes I wrote when I was an undergraduate.

Singular and Supersingular Moduli (pdf)

This is my undergraduate thesis. It 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.

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. 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 (pdf)

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 (pdf)

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