Fibre bundles of rank two on a curve, the determinant bundle and theta functions. (Fibrés de rang deux sur une courbe, fibré déterminant et fonctions thêta. II.) (French) Zbl 0756.14017

Beauville, Arnaud

Bull. Soc. Math. Fr. 119, No. 3, 259-291 (1991).

[For part I see ibid. 116, No. 4, 431-448 (1988; Zbl 0691.14016).]
Let \(C\) be a smooth and complete algebraic curve of genus \(g\) over the field \(\mathbb{C}\) of complex numbers and let \({\mathcal M}_ d\) be the moduli space of semi-stable vector bundles of rank 2, degree \(d\) and fixed determinant. Let \({\mathcal L}\) be the determinant bundle over \({\mathcal M}_ d\). In this paper the author constructs two natural homomorphisms \(\varphi^*_ 0:S^ 2V\to H^ 0({\mathcal M}_ 0,{\mathcal L}^ 2)\), \(\varphi^*_ p:\bigwedge^ 2V\to H^ 0({\mathcal M}_ 1,{\mathcal L})\) (the second depending on a point \(p\in C)\), where \(V\) is the vector space of second-order theta functions on the Jacobian of \(C\). These homomorphisms are in general isomorphisms. The author constructs in this way bases \(\{d_ k\}\) for the vector spaces \(H^ 0({\mathcal M}_ 0,{\mathcal L}^ 2)\) and \(H^ 0({\mathcal M}_ 1,{\mathcal L})\) in terms of the theta- characteristics of the curve. These results are based on the study of the Kummer variety of \(C\) and the study of the action of the Heisenberg group on \(S^ 2V\) and \(\bigwedge^ 2V\).

Reviewer: J.M.Muñoz Porras (Salamanca)

MSC:

14H42	Theta functions and curves; Schottky problem
14M12	Determinantal varieties
14K25	Theta functions and abelian varieties
14H10	Families, moduli of curves (algebraic)

Keywords:

moduli space; determinant bundle; second-order theta function on the Jacobian; Kummer variety

Citations:

Zbl 0691.14016

References:

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