Theta divisors and the geometry of tautological model.

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 > 2. In this paper we show a relation between theta divisor associated to E and the geometry of the tautological model 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, its theta divisor defines an irreducible component of the variety parametrizing (g-2)-linear spaces which are g-secant the tautological model of 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