Research


My research is mainly concerned with the study of curves on algebraic varieties and their moduli spaces. The questions I think about typically have a classical geometric flavor. I like to use ad hoc specialization/degeneration techniques to attack such problems. Ocasionally, I borrow basic ideas or constructions from other related areas, especially Gromov-Witten theory.

Papers

5. Reducible Calabi-Yau threefolds with countably many rational curves
8 pages, Winter '18
preprint (pdf, arXiv)
Description. It is a reasonable expectation that all rational curves on "most" Calabi-Yau threefolds are isolated. To the best of my knowledge, this isn't actually known in even a single case (barring some vacuous cases when rational curves don't exist). In this paper I give a class of examples of "reducible CYs" which satisfy a suitably defined analogous property. Unfortunately, I don't know whether any of these examples are smoothable and suspect they are probably not. This is a spin-off from paper no. 2 below.

4. The irreducibility of the generalized Severi varieties
32 pages, Fall '17
submitted (pdf, arXiv)
Description. I prove that the generalized Severi varieties introduced by Caporaso and Harris are irreducible (or rather a slight variant of them, since we need to insist that the curves are irreducible). The subjective purpose of the paper is to give an alternative proof of the irreducibility of the usual Severi varieties which bypasses the reduction to the genus 0 case. This is motivated by potential generalizations to irrational surfaces.

3. Elliptic surfaces and linear systems with fat points
12 pages, Fall '17
submitted (pdf, arXiv)
Description. This short note is concerned with some particular instances of the following well-known problem: when do linear systems on surfaces with imposed multiplicities at several general points have the expected dimension? The main observation is that for the "Atiyah ruled surface" with a particular polarization, the special case of one fat point implies the general case of arbitrarily many fat points, as well as results concerning other surfaces, including K3s. I conjecture that this one-point case is true in characteristic 0, but show that it's false in any positive characteristic.

2. Deformation of quintic threefolds to the chordal variety
20 pages, Fall '15
Journal: Transactions AMS, to appear (pdf, arXiv)
Description. I construct a degeneration of quintic threefolds, for which it is possible to describe explicitly all the "admissible" genus 0 stable maps to the degenerate threefold. There is a partial extension to positive genera, which allows us to prove the existence of rigid (isolated) curves of arbitrary genus g on a generic quintic threefold. This was previously known only for g up to 22.

1. Rational curves on del Pezzo manifolds
17 pages, Fall '15
Journal: Advances in Geometry, to appear (pdf, arXiv)
Description. I use a specialization trick to prove a particular, but remarkably simple, enumerative identity. The formula relates the "genus 0, point-conditions-only" GW invariant(s) of a Fano 3-fold of index 2 to the similar invariants of a del Pezzo surface of the same degree as the threefold. Along the way, we do some intersection theory on a quotient of an abelian variety (a power of an elliptic curve) by a classical Weyl group.

0. Degenerations, log K3 pairs and low genus curves on algebraic varieties
PhD Thesis, Harvard University
Description. Consists of papers no. 1 and 2 above and another piece which I don't currently (ever?) intend to publish.


Supporting files for 2. The combinatorics involved in constructing the degenerate curves which smooth out to rigid curves is quite complicated and relies on some ad hoc "building blocks." The validity of these examples can be verified, which suffices logically to complete the proof of the main theorem in the paper, using this simple C/C++ code. The code used to find them is this, but it is unfortunately completely unreadable.