Related papers: Tautological equations in genus 2 via invariance c…
The aim of Part II is to explore the technique of invariance of tautological equations in the realm of Gromov--Witten theory. The main result is a proof of Invariance Theorem (Invariance Conjecture~1 in [14]), via the techniques from…
The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the framework gives an efficient algorithm to calculate all tautological equations using only finite dimensional…
We show an invariance result for the L2-torsion of groups under uniform measure equivalence provided a measure-theoretic version of the determinant conjecture holds. The measure-theoretic determinant conjecture is discussed and, for…
Based on computational evidence, we formulate a number of conjectures on the distribution of rational points on curves of genus 2 over the rational numbers, in terms of the size of the coefficients of an equation of the form y^2 = f(x) >.
We give a simple combinatorial proof of the $\lambda_g$ conjectue in genus 2. We use a description of the class $\lambda_2$ as a linear combination of boundary strata, and show the conjecture follows inductively from applications of the…
In this paper, we study relations among known universal equations for Gromov-Witten invariants at genus 1 and 2.
We calculate the genus 2 correlation functions of two-dimensional topological gravity, in a background with two primary fields, using the genus 2 topological recursion relations.
We prove that the Tate conjecture is invariant under Homological Projective Duality (=HPD). As an application, we prove the Tate conjecture in the new cases of linear sections of determinantal varieties, and also in the cases of complete…
In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for…
We find explicit equations for two-coverings of Jacobians of genus two curves over an arbitrary ground field of characteristic different from two.
We prove the $p$-curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological arguments.
The main goal of this paper is to prove the following two conjectures for genus up to two: 1. Witten's conjecture on the relations between higher spin curves and Gelfand--Dickey hierarchy. 2. Virasoro conjecture for target manifolds with…
Assuming natural variational realization conjectures, we give uniform bounds for the obstruction to the integral Tate conjecture in 1-dimensional families of algebraic varieties over an infinite finitely generated field.
The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning…
In this article we study the (cohomological) Hodge conjecture for singular varieties. We prove the conjecture for simple normal crossing varieties that can be embedded in a family where the Mumford-Tate group remains constant. We show how…
We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of…
A new tautological equation of $\Mbar_{3,1}$ in codimension 3 is derived and proved, using the invariance condition explained in earlier works.
We consider some string invariants at genus two that appear in the analysis of the $D^8\mathcal{R}^4$ and $D^6\mathcal{R}^5$ interactions in type II string theory. We conjecture a Poisson equation involving them and the Kawazumi--Zhang…
We observe that a lemma used in the study of even sets of nodes on surfaces applies almost verbatim to prove a celebrated formula of Gauss on the 2-torsion of the class group of a quadratic field.
A two-dimensional nonlinear gauge theory that can be proposed for generalization to higher dimensions is derived by means of cohomological arguments.