I have broad interests in classical algebraic geometry. I spend a lot of time thinking about moduli spaces of curves and related spaces, especially their divisor theory. I'm currently working towards understanding the large genus, degree limits of the Hurwitz space motive. My collaborators in these areas are: Gabriel Bujokas, Dawei Chen, Anand Deopurkar, and Ravi Vakil.
In recent work with Aaron Landesman, I've turned my attention to exploring various interpolation problems in projective space. Some interesting enumerative problems have come out of this project, and I plan to pursue them in the near future.
Here are my papers.
- Vector bundles and finite covers (with Anand Deopurkar)
(pdf) (arXiv) (Submitted, 2016)
We study the problem of determining which vector bundles arise as Tschirnhausen bundles of finite maps. In particular, we prove that every vector bundle on a smooth curve, up to twisting by line bundles, is a Tschirnhausen bundle. As a corollary of our main theorem, we deduce that a general branched cover (of any curve of genus at least two) has a stable Tschirnhausen bundle, provided the number of branch points is large.
- Aaron Landesman's senior thesis: Interpolation in Algebraic Geometry
- Moduli of linear sections of a general hypersurface
We consider the basic question: Given a general degree d hypersurface X in projective space Pr, what can one say about the global variation of the moduli of m-plane sections of X? We prove: If d > 2r+1, then the map to the moduli space of degree d hypersurfaces in Pm is generically injective for all m. We also study the monodromy action on the set of lines meeting X at the minimal number of points, generalizing a result about plane curves due to Harris. Finally we solve the following enumerative problem, generalizing a result of Cadman and Laza: Suppose X is a general hypersurface as above, and suppose d=2r+1. How many lines meet X with the same cross-ratios?
- Syzygy divisors on Hurwitz spaces (with Anand Deopurkar)
We investigate the geometry of the loci where syzygy bundles of branched covers are unbalanced. We do several divisor class computations on a partial compactification of Hurwitz space.
- Interpolation Problems: del Pezzo Surfaces (with Aaron Landesman)(pdf)(arXiv)(Submitted, 2016).
We study "interpolation problems" which ask: When does a particular class of projective varieties pass through the expected number of points in projective space? In particular, we show that del Pezzo surfaces indeed do interpolate through the expected number of general points in projective space. We also relate the number of 3-Veronese surfaces through thirteen general points in P9 to a tractable (yet still challenging) enumerative problem in the Hilbert scheme of three points in the plane.
- Slope bounds for families of covers of an elliptic curve (with Gabriel Bujokas)(Coming soon).
We study analogues of the Maroni and Casnati-Ekedahl divisors in the context of branched covers of elliptic curves. We find an intriguing pattern in the slope of an edge of the effective cone of divisors in the lambda-delta plane.
- Special codimension one loci in Hurwitz spaces (pdf)(arXiv)(Submitted, 2015).
We study the Maroni and Casnati-Ekedahl divisors on Hurwitz spaces. We show that these divisors span extremal rays in the effective cone when the degree is at most five. We then show how these divisors provide sharp slope bounds for sweeping families of trigonal, tetragonal, and pentagonal curves. In the process, we prove that the classes of boundary components in the admissible covers compactification are independent, for all degrees and genera. (This paper is an expansion of my thesis.)
- On the Chow ring of the Hurwitz space of degree 3 covers of P1 (with Ravi Vakil) (pdf)(arXiv)(Submitted, 2015).
We compute the Chow ring of the space of trigonal curves. In particular, we show that the Chow ring is tautological, generated by kappa classes.
- Invariants of a general cover of P1 (with Gabriel Bujokas)(pdf)(arXiv)(Submitted, 2015).
We prove that a general cover of the projective line has a balanced "bundle of quadrics" F. We translate the problem into a maximal rank problem for "links" of rational curves in projective space.
- Extremal effective divisors of Brill-Noether and
Gieseker-Petri type in M1,n (with Dawei Chen)(pdf)(arXiv)
Advances in Geometry. (2014)
We show that the Gieseker-Petri divisor in genus 4 and the d-gonal divisors in odd genera pull back to extremal divisors in M1,n under a particular gluing map. The resulting extremal divisors are shown to be different from the (infinitely many) Chen-Coskun extremal divisors.
- The Picard rank conjecture for the Hurwitz spaces of
degree up to five (with Anand Deopurkar)
Algebra and Number Theory (2015)
We show that the Picard groups of the Hurwitz spaces parametrizing simply branched covers of the projective line of degree up to five are torsion. We also investigate the geometry of the Maroni stratification of Hurwitz spaces by realizing some strata as Severi varieties on Hirzebruch surfaces.
- Sharp slope bounds for sweeping families of trigonal curves (with Anand Deopurkar) (pdf)(arXiv)
Math. Res. Lett. (2013)
We prove that 7+20/(3g+1) is a sharp upper bound for the slope of a sweeping curve in the locus of trigonal curves of odd genus. The even genus case was originally settled by Z. Stankova.
- The Geometry of Hurwitz Space (thesis)(2013)
We study the stratification of Hurwitz space by the Maroni and Casnati-Ekedahl loci. As a result, we provide sharp slope bounds for sweeping families in the tetragonal and pentagonal loci, where the genus is subject to a congruence relation with the degree. We also prove the independence of the boundary components of the admissible covers compactification.