Orbit parametrizations of theta characteristics on hypersurfaces over arbitrary fields

Abstract

It is well-known that theta characteristics on smooth plane curves over a field of characteristic different from two are in bijection with certain smooth complete intersections of three quadrics. We generalize this bijection to possibly singular hypersurfaces of any dimension over arbitrary fields including those of characteristic two. It is accomplished in terms of linear orbits of tuples of symmetric matrices instead of smooth complete intersections of quadrics. As an application of our methods, we give a description of the projective automorphism groups of complete intersections of quadrics generalizing Beauville's results.

arXiv comment35 pages

References in corpus (5)

• Determinantal hypersurfaces
• Maximal linear spaces contained in the base loci of pencils of quadrics
• The local-global principle for symmetric determinantal representations of smooth plane curves
• Complete Intersections of Two Quadrics and Galois Cohomology
• The local-global principle for symmetric determinantal representations of smooth plane curves in characteristic two

Cited by in corpus (5)

• A positive proportion of cubic curves over Q admit linear determinantal representations
• The local-global principle for symmetric determinantal representations of smooth plane curves
• The local-global principle for symmetric determinantal representations of smooth plane curves in characteristic two
• Sheaf theoretic classifications of pairs of square matrices over arbitrary fields
• Linear determinantal representations of smooth plane cubics over finite fields