Abstract
Let E be a stable vector bundle of rank r and slope \(2g-1\) on a smooth irreducible complex projective curve C of genus \(g \ge 3\). In this paper we show a relation between theta divisor \(\Theta _E\) and the geometry of the tautological model \(P_E\) of E. In particular, we prove that for \(r > g-1\), if C is a Petri curve and E is general in its moduli space then \(\Theta _E\) defines an irreducible component of the variety parametrizing \((g-2)\)-linear spaces which are g-secant to the tautological model \(P_E\). Conversely, for a stable, \((g-2)\)-very ample vector bundle E, the existence of an irreducible non special component of dimension \(g-1\) of the above variety implies that E admits theta divisor.
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Acknowledgements
I would like to thank A. Verra for many valuable suggestions.
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Brivio, S. Theta divisors and the geometry of tautological model. Collect. Math. 69, 131–150 (2018). https://doi.org/10.1007/s13348-017-0198-2
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DOI: https://doi.org/10.1007/s13348-017-0198-2