An algorithm to compute fundamental classes of spin components of strata of differentials

TL;DR

This paper presents an algorithm for computing the cycle classes of spin components in strata of differentials within the moduli space of stable curves, implemented in Sage, enabling verification of conjectures on spin double ramification cycles.

Contribution

The paper introduces a novel recursive algorithm for calculating spin component classes and implements it in Sage, facilitating new computational approaches in algebraic geometry.

Findings

Algorithm successfully computes spin class cycles.

Implementation in Sage enhances computational accessibility.

Verifies conjectures on spin double ramification cycles.

Abstract

We construct an algorithm for computing the cycle classes of the spin components of a stratum of differentials in the moduli space of stable curves $\overline{\mathcal{M}}_{g,n}$. In addition, we implement it within the Sage package admcycles. Our main strategy is to reconstruct these cycles by their restrictions to the boundary of $\overline{\mathcal{M}}_{g,n}$ via clutching maps. These restrictions can be computed recursively by smaller dimensional spin classes and determine the original class via a certain system of linear equations. To study the spin parities on the boundary of a stratum of differentials of even type, we make use of the moduli space of multi-scale differentials introduced in [BCCGM19]. As an application of our algorithm, one can verify a conjecture on spin double ramification cycles stated in [CSS21] in many examples, by using the results computed by our algorithm.