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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asked Jun 21, 2013 at 21:34
Mohammad Farajzadeh-TehraniMohammad Farajzadeh-Tehrani
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$\begingroup$ 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. $\endgroup$– Charles StaatsCommented Jun 21, 2013 at 22:33
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2$\begingroup$ Dear Mohammed: These are called "Ulrich bundles" in the literature. There are many references. $\endgroup$– Jason StarrCommented Jun 21, 2013 at 23:38
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1 Answer
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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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$\begingroup$ Is it possible for this line bundle $L$ to satisfy $L\otimes L = K_C$ ($K_C$ is the canonical bundle) as well? $\endgroup$ Commented Jun 21, 2013 at 23:16
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2$\begingroup$ 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. $\endgroup$ Commented Jun 22, 2013 at 5:24
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$\begingroup$ Another name for such $L$ is {\it theta-characteristic}. $\endgroup$– SashaCommented Jun 22, 2013 at 17:39