An Algorithm to Compute the Fundamental Classes of Spin Components of Strata of Differentials

Abstract

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 [4]. As an application of our algorithm, one can verify a conjecture on spin double ramification cycles stated in [18] in many examples, by using the results computed by our algorithm.