Related papers: Computing topological invariants with one and two-…
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…
We give two recursions for computing top intersections of tautological classes on blowups of moduli spaces of genus-one curves. One of these recursions is analogous to the well-known string equation. As shown in previous papers, these…
The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…
We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert…
The theory of Q-Cartier divisors on the space of n-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of Q-Divisors is…
We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and…
We describe algorithms for computing the intersection numbers of divisors and of Chern classes of the Hodge bundle on the moduli spaces of stable pointed curves. We also discuss the implementations and the results obtained. There are…
We generalize two classical formulas for complete intersection curves by introducing the the complete intersection discrepancy of a curve as a correction term. The first is a well-known multiplicity formula in singularity theory, due to…
We investigate the geometry and topology of a standard moduli space of stable bundles on a Riemann surface, and use a generalization of the Verlinde formula to derive results on intersection pairings.
We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points…
At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process…
We describe a new perspective on the intersection theory of the moduli space of curves involving both Virasoro constraints and Gorenstein conditions. The main result of the paper is the computation of a basic 1-point Hodge integral series…
Let $M^m$ be an oriented manifold, let $N^{m-1}$ be an oriented closed manifold, and let $p$ be a point in $M^m$. For a smooth map $f:N^{m-1} \to M^m, p \not\in Im f,$ we introduce an invariant $awin_p(f)$ that can be regarded as a…
This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues' density, and write the universal limit in function of the Kontsevich…
We study moduli spaces of twisted maps to a smooth pair in arbitrary genus, and give geometric explanations for previously known comparisons between orbifold and logarithmic Gromov--Witten invariants. Namely, we study the space of twisted…
We show that the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlev\'e equation, constructed from the Airy functions, as well as associated solutions of P$_\mathrm{XXXIV}$ and S$_\mathrm{II}$, can be expressed…
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ``conformal'' (multicomponent) and Kontsevich models are considered in some detail, together with the…
The main goal of this article is to expand the theory of invariants of Artin-Schreier curves by giving a complete classification in genus 3 and 4. To achieve this goal, we first establish standard forms of Artin-Schreier curves and…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for…