Scaling of conformal blocks and generalized theta functions over $\bar{M}_{g,n}$

TL;DR

This paper investigates the geometric interpretation of conformal blocks over the moduli space of curves, revealing conditions under which these interpretations hold or fail, and exploring related recursion relations for line bundle Chern classes.

Contribution

It demonstrates that geometric interpretations of conformal blocks as sections of line bundles may not extend to boundary points and identifies conditions where these interpretations are valid.

Findings

Recursion relations for Chern classes can fail at boundary points.

Geometric interpretations hold under specific conditions.

Intersection theory provides insights into the structure of conformal blocks.

Abstract

By way of intersection theory on $\bar M_{g,n}$, we show that geometric interpretations for conformal blocks, as sections of ample line bundles over projective varieties, do not have to hold at points on the boundary. We show such a translation would imply certain recursion relations for first Chern classes of these bundles. While recursions can fail, geometric interpretations are shown to hold under certain conditions.