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Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$. By Riemann-Roch theorem, $$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$

Question: For which $g$, there is such $E$ with $H^0(E)=0$ (and thus $H^1(E)=0$ as well)?

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This question is equivalent to asking for which curves there exists a rank-2 vector bundle $E$ such that $H^0(E) = 0 = H^1(E)$. It's pretty clear that $\mathscr{O}(1) \oplus \mathscr{O}(1)$ works for the curve $\mathbb P^1$ (i.e., $g=0$), but I have a feeling the OP already knew this and/or doesn't care about this case. – Charles Staats Jun 21 '13 at 22:33
Dear Mohammed: These are called "Ulrich bundles" in the literature. There are many references. – Jason Starr Jun 21 '13 at 23:38
up vote 5 down vote accepted

Every curve admits a degree g-1 line bundle with $h^0=h^1=0$ -- in fact a generic degree g-1 line bundle has this property, since the space of degree g-1 divisors is g-1 dimensional, but the space of line bundles is g dimensional. Taking the direct sum of two such line bundles will give you a bundle of the type you are seeking.

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Is it possible for this line bundle $L$ to satisfy $L\otimes L = K_C$ ($K_C$ is the canonical bundle) as well? – Mohammad F. Tehrani Jun 21 '13 at 23:16
Yes. A line bundle and an isomorphism $L^{\otimes 2} \cong K_C$ is called a spin structure on $C$. The moduli space of pairs ($C$, spin structure on $C$) has two components since one can show that the quantity $h^0(L)\pmod 2$ is constant in families. In the moduli space of "even" spin curves, the ones where $L$ is effective form a divisor. So a general even spin curve gives an example. – Dan Petersen Jun 22 '13 at 5:24
Another name for such $L$ is {\it theta-characteristic}. – Sasha Jun 22 '13 at 17:39

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