Research
Research themes:
moduli spaces of curves and vector bundles, rational curves on hypersurfaces, Brill--Noether theory, Schubert calculus, singularities, arithmetic geometry, nonarchimedean and tropical approaches, matroids.
My ORCID and Google Scholar pages.
Journal articles/preprints:
Clemens conjectured that there are only finitely many rational curves of each degree on a general quintic threefold F in P^4, and that the only singular rational curves on F are six-nodal plane quintics. In this paper, we prove the “strong form” of the Clemens conjecture in degree 10, i.e. we prove the finiteness statement and that each rational curve on F of degree 10 has normal bundle O(-1)^2.
2. Rational curves of degree 11 on a general quintic threefold (arXiv version).
Abbreviated version, published in Quart. J. Math. 63 (2012), no. 3, 539--568.
We prove the strong form of Clemens' conjecture in degree 11.
We study linear series on a general curve of genus g, whose images are exceptional with regard to their secant planes. Via an extension of Brill-Noether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes whenever the total number of series is finite. We also partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes.
4. Effective divisors on M_g-bar associated to curves with exceptional secant planes (arXiv version).
Abbreviated version published in Manuscripta Math. 138 (2012), no. 1-2, 171--202.
This paper is a sequel to the preceding one. Here we study effective divisors on the moduli space M_g-bar swept out by curves with secant-exceptional linear series. We describe a strategy for computing the classes of those divisors, and we pay special attention to the extremal case of (2d−1)-dimensional series with d-secant (d−2)-planes, which appears in the study of Hilbert schemes of points on surfaces. In that case, modulo a combinatorial conjecture, we obtain hypergeometric expressions for tautological coefficients that enable us to deduce the asymptotics in d of our divisors' virtual slopes.
In this (mostly) survey article, we give a synopsis of a number of results relating to Brill--Noether theory on curves and metric graphs, together with some speculations about the behavior of one-dimensional linear series on a class of metric graphs that admit decompositions as triples of trees rooted on a common vertex set.
On the basis of dimension heuristic, one expects that no rational curves lie on a general hypersurface of degree 7 in P^5. We show that the expectation holds for rational curves in P^5 of degree 16, thereby extending a result of Hana and Johnsen, who had proved the corresponding statement in degrees at most 15.
Here we address the validity of a natural (co)dimension heuristic for singular rational curves in P^n of degree d and arithmetic genus g suggested by our earlier work on Clemens' conjecture. Let M^n_d (resp., M^n_{d,g}) denote the space of morphisms from P^1 to P^n of degree d (resp., of degree d and with image of arithmetic genus g). Whenever d >> g, we expect M^n_{d,g} to be of codimension (n-2)g inside of M^n_d. In this paper we verify this "(n-2)g conjecture" for small values of g, by leveraging the arithmetic properties of value semigroups of singularities.
We study rational curves with a unique unibranch genus-g singularity of γ-hyperelliptic type in the sense of Torres; we focus on the cases γ=0 and γ=1, in which the semigroup associated to the singularity is of (sub)maximal weight. We obtain a partial classification of these curves according to the linear series they support, the scrolls on which they lie, and their gonality.
Given a real hyperelliptic algebraic curve X with non-empty real part and a real effective divisor D arising via pullback from P^1 under the hyperelliptic structure map, we study the real inflection points of the associated complete real linear series $|D|$ on X. To do so we use Viro's patchworking of real plane curves, recast in the context of some Berkovich spaces studied by M. Jonsson. Our method gives a simpler and more explicit alternative to limit linear series on metrized complexes of curves, as developed by O. Amini and M. Baker, for curves embedded in toric surfaces.
10. Real inflection points of real linear series on an elliptic curve, joint with Cristhian Garay López, Experimental Mathematics (2019), DOI: https://doi.org/10.1080/10586458.2019.1655815.
Given a real elliptic curve E with non-empty real part, we study the real inflection points of distinguished subseries of the complete real linear series determined by kD for k≥3, where D is the g^1_2 on E. We define key (inflection) polynomials whose roots index the (x-coordinates of) inflection points of the linear series, away from the points where E ramifies over P^1. These fit into a recursive hierarchy, in the same way that division polynomials index torsion points. Our study is motivated by, and complements, an analysis of how inflectionary loci vary in the degeneration of real hyperelliptic curves to a metrized complex of curves with elliptic curve components that we carried out in our previous article with Biswas.
In this note, we show that γ-hyperelliptic numerical semigroups of genus g≫γ satisfy a refinement of a well-known characteristic weight inequality due to Torres. The refinement arises from substituting the usual notion of weight by an alternative version, the K-weight, which we previously introduced in the course of our study of unibranch curve singularities.
This is a complement to the study initiated in papers 3 and 4 above. Here we study secant-exceptional linear series on a general curve of genus g. Each exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear series with base points over families of curves of compact type, which we then use to compute combinatorial formulas for the number of secant-exceptional linear series when the spaces of linear series and of inclusions are finite.
This largely-expository note complements papers 8 and 9 above. Inflection divisors on elliptic curves index the inflection points of linear series arising (as subspaces of holomorphic sections) from line bundles on ℙ^1 via pullback along the canonical 2-to-1 projection. Associated to each inflection divisor on an elliptic curve E_λ:y^2=x(x−1)(x−λ), there is an associated inflectionary curve in (the projective compactification of) the affine plane in coordinates x and λ. These inflectionary curves have remarkable features; among other things, they lead directly to an explicit conjecture for the number of real inflection points of linear series on E_λ whenever the Legendre parameter λ is real.
By viewing a rank two vector bundle as an extension of line bundles we may re- interpret cohomological conditions on the vector bundles (e.g., number of sections) as rank conditions on multiplication maps of sections of line bundles. In this paper, we apply this philosophy to relate the Brill–Noether theory of rank two vector bundles with canonical determinant to the Strong Maximal Rank Conjecture for quadrics. By verifying that certain “special maximal-rank loci” are nonempty, we are able to produce candidates for rank two linear series of large dimension. We then show that the underlying vector bundles are stable, in order to conclude the existence portion of certain instances of a well-known conjecture due to Bertram, Feinberg and independently Mukai.
(N,γ)-hyperelliptic semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated to totally-ramified points of N-fold covers of curves of genus γ. Torres characterized (2,γ)-hyperelliptic semigroups of maximal weight whenever their genus is large relative to γ. Here we do the same for (3,γ)-hyperelliptic semigroups, and we formulate a conjecture about the general case whenever N≥3 is prime.
We prove that the variety of (parameterizations of) rational curves of sufficiently large fixed degree d in P^n with a single hyperelliptic cusp of delta-invariant g is always of codimension at least (n−1)g inside the space of degree-d holomorphic maps P^1→P^n; and that when g is small, this bound is exact and the corresponding space of maps is paved by unirational strata indexed by fixed ramification profiles. We give a conjectural generalization of this picture for rational curves with cusps of arbitrary value semigroup S, and provide evidence for this conjecture whenever S is a γ-hyperelliptic semigroup of either minimal or maximal weight. Finally, we produce infinitely many new examples of reducible Severi varieties M^n_{d,g} of holomorphic maps P^1→P^n with images of degree d and arithmetic genus g, for every value of n>2.
Over the complex numbers, Plücker’s formula computes the number of inflection points of a linear series of projective dimension r and degree d on a curve of genus g. Here we explore the geometric meaning of a natural analogue of Plücker’s formula in A^1 -homotopy theory for certain linear series on hyperelliptic curves defined over an arbitrary field.
The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-archimedean (e.g., p-adic) settings. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. Finally, the methods we have used in formulating and proving the fundamental theorem reveal new examples of non-classical valuations that merit further study in their own right.
19. Arithmetic inflection of superelliptic curves, joint with Ignacio Darago, Cristhian Garay López, Changho Han, and Tony Shaska (submitted)
We explore the inflectionary behavior of linear series on superelliptic curves X over fields of arbitrary characteristic. Here we give a precise description of the inflection of linear series over the ramification locus of the superelliptic projection; and we initiate a study of those inflectionary varieties that parameterize the inflection points of linear series on X supported away from the superelliptic ramification locus that is predicated on the behavior of their Newton polytopes.
The Weierstrass semigroup of pole orders of meromorphic functions in a point p of a smooth algebraic curve C is a classical object of study; a celebrated problem of Hurwitz is to characterize which numerical semigroups are realizable as Weierstrass semigroups S= S(C,p). In this note, we establish realizability results for cyclic covers (C,p) -> (B,q) of hyperelliptic targets B marked in hyperelliptic Weierstrass points; and we show that realizability is dictated by the behavior under j-fold multiplication of certain divisor classes in hyperelliptic Jacobians naturally associated to our cyclic covers, as j ranges over all natural numbers.
We prove that an adjusted version of the dimensionality conjecture for Severi varieties of holomorphic maps P^1 -> P^n with unicuspidal images in paper #16 above holds when the underlying cusps are of either hyperelliptic or supersymmetric type.
22. Towards Brill--Noether theory for unicuspidal curves, with Renato Vidal Martins (Matemática Contemporânea 60 (2024), 31--48)
In this largely-expository note, we compile and contextualize the results of papers 7, 8, 16, and 21 above; and we describe a complementary line of inquiry based on the geometry of canonical models of singular curves, with a focus on gonality.
23. Cusps in C^3 with prescribed ramification, with Nathan Kaplan and Renata Vieira Costa (submitted)
We study value semigroups associated to germs of maps C->C^3 with fixed ramification profiles in a distinguished point. We then apply our analysis to deduce that Severi varieties of unicuspidal rational fixed-degree curves with value semigroup S in P^3 are often reducible when S is either 1) the semigroup of a generic cusp whose ramification profile is either a triple of successive multiples of a fixed integer or is a supersymmetric triple; or 2) a supersymmetric semigroup with ramification profile given by a supersymmetric triple. In doing so, we uncover new connections with additive combinatorics and number theory.
24. Matroids and semirings attached to toric singularity arrangements, with Cristhian Garay López
One natural problem is to explicitly determine the value semirings of distinguished infinite classes of singularities, with a view to understanding their asymptotic properties. In this paper, we establish a matroidal framework for resolving this problem for singularities determined by arrangements of toric branches; and we obtain precise quantitative results in the case of line arrangements. We apply our results to produce new examples of unexpectedly large Severi varieties of unisingular rational curves.
0. Geometry of curves with exceptional secant planes (Harvard Ph.D thesis, 2007)
The content of the thesis is covered in papers 3 and 4 above.
Expository
a. Tropical geometry; appeared in MSRI Emissary, January 2010
b. Clemens' conjecture; appeared in Springer online Encyclopedia of Mathematics, August 2012
Other:
Slides from MIT Combinatorics Seminar talk about my Ph.D thesis, October 2007
Poster about "Singular rational curves in P^n via semigroups", Guanajuato, March 2016
Video: "Singular rational curves in P^n via semigroups": Moduli spaces and enumerative geometry, Impa, May 2015 and 14th Alga Meeting, Impa, February 2017
"Rational curves with hyperelliptic singularities": IX Summer workshop on algebraic theory of singularities, Impa, March 2020
Poster about “Real inflection points of real hyperelliptic curves”, AGNES, September 2018
Collaborators with a web presence: Changho Han, Xiang He, César Lozano Huerta, Nathan Kaplan, Cristhian Garay López, Renato Vidal Martins, Nathan Pflueger, Tony Shaska and Naizhen Zhang
Brazilian Algebraic Geometry Seminar (Zoom)
Latin American Real and Tropical Geometry Seminar (Zoom)