A properness result for degenerate Quadratic and Symplectic Bundles on a smooth projective curve

TL;DR

This paper constructs a coarse moduli space for degenerate quadratic and symplectic bundles on smooth projective curves and proves their semi-stable reduction and properness, especially for higher degeneracy cases.

Contribution

It introduces a new moduli construction for degenerate quadratic and symplectic bundles with fixed degeneracy loci and establishes their semi-stable reduction and properness without Bruhat-Tits theory.

Findings

Constructed coarse moduli space for degenerate bundles

Proved semi-stable reduction theorem for these objects

Established properness of polystable orthogonal bundles

Abstract

Let $(V,q)$ be a vector bundle on a smooth projective curve $X$ together with a quadratic form $q: \mathrm{Sym}^2(V) \ra \mathcal{O}_X$ (respectively symplectic form $q: \Lambda^2V \ra \mathcal{O}_X$). Fixing the degeneracy locus of the quadratic form induced on $V/\ker(q)$, we construct a coarse moduli of such objects. Further, we prove semi-stable reduction theorem for equivalence classes of such objects. In particular, the case when degeneracies of $q$ are higher than one is that of principal interest. We also provide a proof of properness of polystable orthogonal bundles without appealing to Bruhat-Tits theory in any characteristic.