Polarisations of type \((1,2,\dots ,2)\) on abelian varieties
The algebraic equivalence class of any ample line bundle on an abelian variety \(A\) of dimension \(r\) determines an \(r\)-tuple \(\delta= (\delta_1, \delta_2, \dots, \delta_r)\) of positive integers with \(\delta_1 |\delta_2 |\cdots |\delta_r\). This is called the type of the line bundle. One might wish to understand at least the generic behaviour of the linear system of \(\tau\), where \(\tau\) is a line bundle of type \(\delta\) on an abelian variety. We are primarily interested here in the case \(\delta =(1,2, \dots, 2)\). First, in the case, there is a base locus consisting of \(2^{2(r-1)}\) points. The linear system gives a morphism of the variety \(\sigma A\) obtained by blowing up these points. Moreover, it is invariant under the natural involution \(\iota\) (upon a proper choice of the line bundle in the algebraic equivalence class). If \(r\geq 4\), we show that the linear system imbeds \(\sigma A/ \iota\) at least birationally. We prove this by considering abelian varieties arising from the following construction. Let \(C\) be a (smooth projective) curve and \(\widetilde C\) a 2-sheeted covering of \(C\) ramified at four points. Then the Prym variety \(P\) is an abelian variety with a polarisation of type \((1,2, \dots, 2)\). We study the rational map given rise to by this linear system by bringing into play the moduli of vector bundles on \(C\) of rank 2 and trivial determinant. In this case (assuming that \(C\) is of genus at least 4), we show that the variety \(\sigma (P)/ \iota\) is imbedded birationally but that there is actually a surface over which the mapping is \((2:1)\). Moreover, the data consisting of \(\widetilde C\), \(C\), etc. can be recovered from the geometry of this imbedding, thus providing a Torelli-type theorem for these Pryms. -- By an obvious degeneracy argument, this proves the result for generic abelian varieties claimed above. However, it is possible that the linear system behaves even better in the case of generic abelian varieties since the varieties that we have considered above are quite special. In \S 8 and \S 9, we consider the case when \(C\) is of genus 3. In fact, we study the general case when, in the above set-up, the branch locus is a divisor which moves in a pencil. Then the special locus consists of three surfaces with properties similar to the above, and hence \(\widetilde C\), \(C\), etc. can be recovered only with an ambiguity of 3. Thus there are three such pairs, and these are related to each other by an analogue of the ``tetragonal construction'' of \textit{R. Donagi} [see Bull. Am. Math. Soc., New Ser. 4, 181-185 (1981; Zbl 0491.14016)]. In view of this, we have clarified in \S 9, and in a sense generalised, Donagi's construction.
scientific article; zbMATH DE number 850804
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