Incidence varieties in the projectivized $k$-th Hodge bundle over curves with rational tails

TL;DR

This paper computes the classes of incidence loci in the projectivized $k$-th Hodge bundle over curves with rational tails, expressing them via tautological classes and providing new relations in tautological rings.

Contribution

It provides explicit formulas for the classes of incidence loci in the moduli space of curves with rational tails, advancing understanding of tautological rings.

Findings

Classes of incidence loci are expressed as linear combinations of tautological classes.

Explicit relations in tautological rings are derived.

Coefficients are given by enumerations over decorated stable graphs.

Abstract

Over the moduli space of pointed smooth algebraic curves, the projectivized $k$-th Hodge bundle is the space of $k$-canonical divisors. The incidence loci are defined by requiring the $k$-canonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized $k$-th Hodge bundle over the moduli space of curves with rational tails. The classes are expressed as a linear combination of tautological classes indexed by decorated stable graphs with coefficients enumerating appropriate weightings. As a consequence, we obtain an explicit expression for some relations in tautological rings of moduli of curves with rational tails.