On Faltings parabolic theta functions

Abstract

In this article, we prove that the Faltings theta functions on moduli of parabolic bundles over a smooth projective curve can be used to give an explicit scheme-theoretic projective embedding.

Quiver representations and their applications

“…More precisely, if F * is a parabolic bundle on C, then there is a line bundle D(E * , F * , V * ) on S defined as the determinant line bundle of a 2-term complex P ; • : P 0 p −→ P 1 of vector bundles on S which computes the cohomology of (E * ⊗ F * ⊗ V * ) 0 locally over S. If χ((E * ⊗ F * ⊗ V * ) 0 ) = 0, there is a section θ F * on S which can be locally identified with det p over S (cf. §4.1, [3,5]). For s ∈ S, using Theorem 4.12, it follows that if θ F * (s) 6 = 0, then the parabolic bundle E * s is semistable.…”

“…In [3], using the isomorphism ψ in (4.10), it is proved that the determinant theta functions on M ss e C (τ ) (see Theorem 4.10) coincide with certain Faltings parabolic theta functions which gives an implicit construction of the moduli space M ss C (τ p ). Consequently, it follows that the the morphism Θ F * : M ss C (τ p ) −→ P N is a closed scheme-theoretic embedding (see [3] for more details).…”

“…In [5], a GIT-free construction of the moduli space of semistable parabolic bundles over a smooth projective curve is constructed using the analogous Faltings parabolic theta functions. In [3], it is proved that Faltings parabolic theta functions can be used to give a closed scheme-theoretic embedding of moduli of semistable parabolic bundles, using the results of [2].…”

“…In this article, our main aim is to survey the results of the papers [13,40,18,1,38,5,2,3] by outlining the ideas rather than reproducing the formal proofs. The literature on the topics discussed here is enormous and the list of references given at the end of this article is by no means complete, the author wishes to apologize for the unintentional exclusions of missing relevant references.…”