# Theta functions for Holomorphic triples

## TL;DR

This paper generalizes theta functions to holomorphic triples on curves, linking semistability to the existence of orthogonal triples, and constructs globally generated theta line bundles on their moduli spaces.

## Contribution

It introduces a new theta divisor concept for holomorphic triples and establishes a criterion for semistability via orthogonal triples, advancing the geometric understanding of these objects.

## Key findings

- Characterization of semistability through orthogonal triples
- Construction of globally generated theta line bundles
- Extension of theta divisor theory to holomorphic triples

## Abstract

We introduce an generalization of the theta divisor to the theory of holomorphic triples on a smooth projective curve $X$. We show that a given triple $T=(E_1 \to E_0)$ is $\alpha$-semistable iff there exists an orthogonal tripe $S=(F_1 \to F_0)$ with given numerical invariants. This yields globally generated theta line bundles on the moduli space of semistable triples.