Factorization of point configurations, cyclic covers, and conformal blocks. (English) Zbl 1329.14060

This paper studies configuration spaces of points on the projective line and in higher-dimensional projective spaces. The first main result (Theorem 1.1) describes a relation between the invariants of \(n\) ordered points in \(\mathbb{P}^d\) and of points contained in a union of two linear subspaces. By studying the projective embedding induced by these invariants, the second main result (Theorem 1.2) describes an attaching map for GIT quotients of point configurations in these spaces that respects the Segre product of the natural GIT polarizations. Associate a cyclic cover to a configuration supported on a rational normal curve. The authors further show that if the branch points are weighted by the GIT linearization and the rational normal curve degenerates, then the admissible covers limit is a cyclic cover with weights in the attaching map. They also find line bundles arising from both GIT polarizations and the Hodge class for families of cyclic covers that have functorial restriction to the boundary. Moreover, the authors introduce an abstraction of Wess-Zumino-Witten factorization to encode the isomorphism class of these line bundles (Proposition 1.6). As an application (Theorem 1.7), they obtain a unified geometric proof of some recent results on conformal block bundles.


14H10 Families, moduli of curves (algebraic)
14N20 Configurations and arrangements of linear subspaces
14L24 Geometric invariant theory
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[1] Alexeev, V., Gibney, A., Swinarski, D.: Higher-level sl2 conformal blocks divisors on M0,n. Proc. Edinburgh Math. Soc. 57, 7-30 (2014) · Zbl 1285.14012 · doi:10.1017/S0013091513000941
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