Related papers: Intersecting Psi-classes on tropical M_{0,n}

We compute the intersection cohomology of the moduli spaces $M_{r,d}$ of semistable vector bundles having rank $r$ and degree $d$ over a curve. We do this by relating the Hodge-Deligne polynomial of the intersection cohomology of $M_{r,d}$…

We study intersection theory on the relative Hilbert scheme of a family of nodal-or-smooth curves, over a base of arbitrary dimension. We introduce an additive group called 'discriminant module', generated by diagonal loci, node scrolls,…

The cross-ratio degree problem is about counting rational curves with $n$ marked points satisfying $n-3$ cross-ratio conditions. This problem has a tropical analogue which provides the same number, as shown by a correspondence theorem. In…

In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.

Let X and X' be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles, in the sense of Henning Meyer's graduate thesis, between the tropicalization of the intersection…

We compute the automorphism group $\mathrm{Aut}(\Delta_{g, n})$ for all $g, n \geq 0$ such that $3g - 3 + n > 0$, where $\Delta_{g, n} \subset M_{g, n}^\mathrm{trop}$ is the moduli space of stable $n$-marked tropical curves of genus $g$ and…

We identify the formulas of Buryak and Okounkov for the n-point functions of the intersection numbers of psi-classes on the moduli spaces of curves. This allows us to combine the earlier known results and this one into a principally new…

Under suitable conditions on a family of logarithmic curves, we endow the tropicalization of the family with an affine structure in a neighborhood of the sections in such a way that the tropical $\psi$ classes from \cite{psi-classes} arise…

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of…

We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature (n-1, 1). We define arithmetic cycles on these models and study their intersection behaviour. In…

We prove that a semiring isomorphism between the rational function semifields of two tropical curves induces an expansive map between those tropical curves. This semiring isomorphism and the expansive map respect zeros and poles of rational…

We study the moduli space of smooth complete intersections of two quadrics in $\mathbb{P}^n$ by relating it to the geometry of the singular members of the corresponding pencils. Giving an alternative presentation for the moduli space of…

We define tropical rational function semifields $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ and prove that a tropical curve $\varGamma$ is realized (except for points at infinity) as the congruence variety $V \subset \boldsymbol{R}^n$…

In this paper we produce infinitely many examples of set-theoretic complete intersection monomial curves in $\mathbb{P}^{n+1}$, starting with a set-theoretic complete intersection monomial curve in $\mathbb{P}^{n}$ . In most of the cases…

Witten's top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with r-spin structures. It plays a key role in Witten's conjecture relating to the intersection theory on these moduli spaces. Our…

We use tropical and non-archimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space $Y$. In particular, we are interested in the choices of parameters for which the generic root…

We extend Gubler--K\"unnemann's theory of $\delta$-forms from algebraic varieties to good Berkovich spaces. This is based on the observation that skeletons in such spaces satisfy a tropical balance condition. Our main result is that…

Numerical equivalence of algebraic cycles is defined abstractly by intersection numbers. Classically, for smooth complex proper toric varieties, the quotients by numerical equivalence with rational coefficients can be described…

We study the question of the continuity of slices of currents and explain how it relates to several seemingly unrelated problems in tropical geometry. On the one hand, through this lens, we show that the continuity of superpotentials…

This note is about invariants of moduli spaces of curves. It includes their intersection theory and cohomology. Our main focus in on the distinguished piece containing the so called tautological classes. These are the most natural classes…