arXiv subjects

Jason van Zelm

Publications and source records attributed to Jason van Zelm.

4 recordsLinked to original sources

arXiv2020

admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves

The tautological ring of the moduli space of stable curves has been studied extensively in the last decades. We present a SageMath implementation of many core features of this ring. This includes lists of generators and their products, intersection numbers and verification of tautological relations. Maps between tautological rings induced by functoriality, that is pushforwards and pullbacks under gluing and forgetful maps, are implemented. Furthermore, many interesting cycle classes, such as the double ramification cycles, strata of k-differentials and hyperelliptic or bielliptic cycles are available. In this paper we show how to apply the package, including concrete example computations.

math.AG↗

arXiv2019

Pullbacks of universal Brill-Noether classes via Abel-Jacobi morphisms

Following Mumford and Chiodo, we compute the Chern character of the derived pushforward $\textrm{ch} (R^\bulletπ_\ast\mathscr{O}(\mathsf{D}))$, for $\mathsf D$ an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.

math.AG↗

arXiv2018

Intersections of loci of admissible covers with tautological classes

For a finite group $G$, let $\H_{g,G,ξ}$ be the stack of admissible $G$-covers $C\to D$ of stable curves with ramification data $ξ$, $g(C)=g$ and $g(D)=g'$. There are source and target morphisms $ϕ\colon \H_{g,G,ξ}\to \M_{g,r}$ and $δ\colon \H_{g,G,ξ}\to \M_{g',b}$, remembering the curves $C$ and $D$ together with the ramification or branch points of the cover respectively. In this paper we study admissible cover cycles, i.e. cycles of the form $ϕ_* [\H_{g,G,ξ}]$. Examples include the fundamental classes of the loci of hyperelliptic or bielliptic curves $C$ with marked ramification points. The two main results of this paper are as follows: Firstly, for the gluing morphism $ξ_A\colon \M_A\to \M_{g,r}$ associated to to a stable graph $A$ we give a combinatorial formula for the pullback $ξ^*_A ϕ_*[\H_{g,G,ξ}]$ in terms of spaces of admissible $G$-covers and $ψ$ classes. This allows us to describe the intersection of the cycles $ϕ_*[\H_{g,G,ξ}]$ with tautological classes. Secondly, the pull-push $δ_*ϕ^*$ sends tautological classes to tautological classes and we also give a combinatorial description of this map in terms of standard generators of the tautological rings. We show how to use the pullbacks to algorithmically compute tautological expressions for cycles of the form $ϕ_* [\H_{g,G,ξ}]$. In particular, we compute the classes $[\Hyp_5]$ and $[\Hyp_6]$ of the hyperelliptic loci in $\M_5$ and $\M_6$ and the class $[\B_4]$ of the bielliptic locus in $\M_4$.

math.AG↗

arXiv2016

Nontautological Bielliptic Cycles

Let $[\overline{\mathcal{B}}_{2,0,20}]$ and $[\mathcal{B}_{2,0,20}]$ be the classes of the loci of stable resp. smooth bielliptic curves with 20 marked points where the bielliptic involution acts on the marked points as the permutation (1 2)...(19 20). Graber and Pandharipande proved that these classes are nontatoulogical. In this note we show that their result can be extended to prove that $[\overline{\mathcal{B}}_{g}]$ is nontautological for $g\geq 12$ and that $[\mathcal{B}_{12}]$ is nontautological.

math.AG↗