Publications

Relative Gromov-Witten and maximal contact conic

Submitted - 2024

We discuss some properties of the relative Gromov-Witten invariants counting rational curves with maximal contact order at one point. We compute the number of Cayley's sextactic conics to any smooth plane curve Y. In particular, we compute the contribution, from double covers of inflectional lines, to a certain degree 2 relative Gromov-Witten invariant relative to Y.

Keywords: relative, Gromov-Witten, sextactic, osculating, tangent, rational.

An arithmetic count of osculating lines

Submitted - 2023

We say that a line in P^(n+1)_k is osculating to a hypersurface Y if they meet with contact order n+1. When k=C, it is known that through a fixed point of Y, there are exactly n! of such lines. Under some parity condition on n and deg(Y), we define a quadratically enriched count of these lines over any perfect field k. The count takes values in the Grothendieck-Witt ring of quadratic forms over k and depends linearly on deg(Y).

Keywords: Osculating curves, bilinear form, Grothendieck-Witt.

Localization in Gromov—Witten Theory of Toric Varieties in a Computer Algebra System

Lecture Notes in Computer Science ((volume 14749)) - Conference paper.

Computations of Gromov-Witten invariants of toric varieties

Journal of Symbolic Computation - 2023

We present the Julia package ToricAtiyahBott.jl, providing an easy way to perform the Atiyah-Bott formula on the moduli space of genus 0 stable maps to any smooth projective toric variety. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety, as well as the cohomology class of the curves, must be defined using the package Oscar.jl.

Keywords: torus action, Bott formula, enumeration, Julia programming language, toric varieties.

Irreducible Contact Curves via Graph Stratification

Bulletin des Sciences Mathématiques - 2022

We prove that the moduli space of contact stable maps to any odd dimensional projective space of degree d admits a stratification parameterized by graphs. We use it to determine the number of irreducible rational contact curves with any Schubert condition. We give explicitely some of these invariants for the projective space of dimension 3 and 5.

Keywords: Legendrian, contact, enumeration, symplectic, stable maps, rational curves .

Effective computations of the Atiyah-Bott formula

with Csaba Schneider - Journal of Symbolic Computation - 2021

We present an implementation of the Atiyah-Bott residue formula for the moduli space of rational pointed stable maps to a projective space. We use this implementation to compute a large number of Gromov-Witten invariants of genus 0, including intersection numbers of rational curves on general complete intersections. We also compute some numbers of rational contact curves satisfying suitable Schubert conditions. Our computations confirm known predictions made by Mirror Symmetry. The code we developed for these computations is publicly available and can be used for other types of computations.

Keywords: Legendrian, contact, torus action, Bott formula, enumeration, SageMath, Julia programming language.

Enumeration of rational contact curves via torus actions.

Michigan Mathematical Journal - 2021

We prove that some Gromov-Witten numbers associated to rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative. Also, we use Bott formula on the Kontsevich space to find the exact value of those numbers. As an example, the numbers of rational contact curves of low degree in P3 and P5 are computed. The results are consistent with existing results.

Keywords: Legendrian, contact, torus action, Bott formula, Enumeration.

A Recursive Formula for Osculating Curves.

Arkiv för Matematik - 2020

Let X be a smooth complex projective variety. Using a construction devised by Gathmann, we present a recursive formula for some of the Gromov-Witten invariants of X. We prove that, when X is homogeneous, this formula gives the number of osculating rational curves at a general point of a general hypersurface of X. This generalizes the classical well known pairs of inflection (asymptotic) lines for surfaces in  P3  of Salmon, as well as Darboux's 27 osculating conics.

Keywords: Gromov-Witten invariants, osculating curves, homogeneous varieties.

The indeterminacy locus of the Voisin map.

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry - 2017

Keywords: IHS, hyperkähler, Voisin map.

Betti numbers and pseudoeffective cones in 2-Fano varieties.

Advances in Geometry - 2016

The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties in analogy with the case k=1. Then, we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index greater than n-3, and also we complete the classification of weak 2-Fano varieties of Araujo and Castravet.

Keywords: higher Fano, pseff cones.

Pseudoeffective cones in 2-Fano varieties and remarks on the Voisin map.

My Ph.D. thesis, in joint program between University Roma Tre and Strasbourg.