Kapranov degrees

TL;DR

This paper introduces Kapranov degrees, a new combinatorial invariant of the moduli space of stable rational curves, providing bounds and positivity criteria, and connects these to graph rigidity and error-correcting codes.

Contribution

It defines Kapranov degrees as multidegrees of a certain embedding, establishes bounds and positivity conditions, and links these to graph rigidity and coding theory.

Findings

Established upper bounds for Kapranov degrees.

Provided a combinatorial characterization of their positivity.

Connected positivity to Laman's theorem and graph rigidity.

Abstract

The moduli space of stable rational curves with marked points has two distinguished families of maps: the forgetful maps, given by forgetting some of the markings, and the Kapranov maps, given by complete linear series of $\psi$-classes. The collection of all these maps embeds the moduli space into a product of projective spaces. We call the multidegrees of this embedding ``Kapranov degrees,'' which include as special cases the work of Witten, Silversmith, Gallet--Grasegger--Schicho, Castravet--Tevelev, Postnikov, Cavalieri--Gillespie--Monin, and Gillespie--Griffins--Levinson. We establish, in terms of a combinatorial matching condition, upper bounds for Kapranov degrees and a characterization of their positivity. The positivity characterization answers a question of Silversmith and gives a new proof of Laman's theorem characterizing generically rigid graphs in the plane. We achieve this by proving a recursive formula for Kapranov degrees and by using tools from the theory of error correcting codes.