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Let us study vector bundles $E$ and $F$ on a smooth projective curve $C$. There is a Riemann-Roch type formula for the Euler characteristic $\chi(E,F)=dim\, Hom(E,F)-dim\, Ext^1(E,F)$ in terms of degrees $d(E),d(F)$, ranks $r(E),r(F)$ and the genus $g$ of $C$. Assume that this formula gives a non-positive number (this is equivalent to $\mu(F)-\mu(E)\le g-1$).

Is it true that for generic $E$ and $F$ with such ranks and degrees one has $Hom(E,F)=0$? Or is it true at least for some $E,F$?

(This is clearly true for line bundles, but I failed to answer the question for arbitrary vector bundles after several days of thinking)

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I have found the answer, it is positive. The statement is equivalent to a theorem of Hirschowitz, see Th. 1.2 in arXiv:alg-geom/9710019v2.

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  • $\begingroup$ The paper you mention is published: B. Russo, M. Teixidor, On a conjecture of Lange, J. Algebraic Geom. 8 (1999), no. 3, 483–496. $\endgroup$
    – abx
    Commented Mar 21, 2020 at 6:43

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