Vanishing thetanulls on curves with involutions
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Vanishing thetanulls on curves with involutions Arnaud Beauville
Received: 29 June 2012 / Accepted: 3 October 2012 / Published online: 12 February 2013 © Springer-Verlag Italia 2013
Abstract The configuration of theta characteristics and vanishing thetanulls on a hyperelliptic curve is completely understood. We observe in this note that analogous results hold for the σ -invariant theta characteristics on any curve C with an involution σ . As a consequence we get examples of non hyperelliptic curves with a high number of vanishing thetanulls. Keywords involution
Thetanullwerte · Theta characteristics · Vanishing thetanulls · Curves with
Mathematics Subject Classification
Primary 14H42; Secondary 14H51 · 14H45
1 Introduction Let C be a smooth projective curve over C. A theta characteristic on C is a line bundle κ such that κ 2 ∼ = K C ; it is even or odd according to the parity of h 0 (κ). An even theta characteristic κ with h 0 (κ) > 0 is called a vanishing thetanull. The terminology comes from the classical theory of theta functions. A theta characteristic κ corresponds to a symmetric theta divisor κ on the Jacobian J C, defined by a theta function θκ ; this function is even or odd according to the parity of κ. Thus the numbers θκ (0) are 0 for κ odd; for κ even they are classical invariants attached to the curve (“thetanullwerte” or “thetanulls”). The thetanull θκ (0) vanishes if and only if κ is a vanishing thetanull in the above sense. When C is hyperelliptic, the configuration of its theta characteristics and vanishing thetanulls is completely understood (see e.g. [4]). We observe in this note that analogous results hold for the σ -invariant theta characteristics on any curve C with an involution σ . As a consequence we obtain examples of non hyperelliptic curves with a high number of
A. Beauville (B) Laboratoire J.-A. Dieudonné, UMR 7351 du CNRS, Université de Nice, Parc Valrose, 06108 Nice Cedex 2, France e-mail: [email protected]
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vanishing thetanulls: for instance approximately one fourth of the even thetanulls vanish for a bielliptic curve. 2 σ -Invariant line bundles Throughout the paper we consider a curve C of genus g, with an involution σ . We denote by π : C → B the quotient map, and by R ⊂ C the fixed locus of σ . For a subset E = { p1 , . . . , pk } of R we will still denote by E the divisor p1 + · · · + pk . The double covering π determines a line bundle ρ on B such that ρ 2 = O B (π∗ R); we have π ∗ ρ = OC (R), π∗ OC ∼ = O B ⊕ ρ −1 and K C = π ∗ (K B ⊗ ρ). We consider the map ϕ : Z R → Pic(C) which maps r ∈ R to the class of OC (r ). Its image lies in the subgroup Pic(C)σ of σ -invariant line bundles. Lemma 1 ϕ induces a surjective homomorphism ϕ¯ : (Z/2) R → Pic(C)σ /π ∗ Pic(B), whose kernel is Z/2 · (1, . . . , 1). Proof Let RC and R B be the fields of rational functions of C and B, respectively. Let σ (∼ = Z/2) be the Galois group of the covering π. Consider the exact sequence of σ -modules 1 → RC∗ /C∗ → Div(C) → Pic(C) → 0 . Since H 1 ( σ , RC∗ ) = 0 by Hi
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