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Theta characteristics in trigonal curves. (English) Zbl 1096.14023

Let \(C\) be a trigonal curve of genus \(g\geq 5\). Let \(m\) and \(n\) be the invariants of \(C\). The authors show:
(1) If \(m<2r\leq n\), then \(C\) has a theta characteristic of type \(g^{r}_{g-1}\) if and only if the exceptional divisor can be written as \(E=2F+\sum Q\), where the last summation has \(n-2r\) distinct points \(Q\not\in \text{ supp}(F)\) such that the trigonal divisor over \(Q\) is ramified;
(2) If \(n-m-1>r\), then \(C\) does not admit a theta characteristic of dimension \(r\);
(3) If \(n-m-1\leq r\), then the dimension of the moduli space of trigonal curves of genus \(g\) (with invariants \(m\) and \(n\)) and having a theta characteristic of dimension \(r\) is \(g+m+r+3\).

MSC:

14H45 Special algebraic curves and curves of low genus
14H05 Algebraic functions and function fields in algebraic geometry
14H51 Special divisors on curves (gonality, Brill-Noether theory)
14C20 Divisors, linear systems, invertible sheaves
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References:

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