Abstract
We describe the image of the locus of hyperelliptic curves of genus g under the period mapping in a neighbourhood of the diagonal locus Diag g . There is just one branch for each of the alkanes C g H 2 g + 2 of elementary organic chemistry, and each branch has a simple linear description in terms of the entries of the period matrix. This picture is replicated for simply connected Jacobian elliptic surfaces, which form the next simplest class of algebraic surfaces after K3 and abelian surfaces. In the period domain for such surfaces of geometric genus g, there is a locus W 1 g that is analogous to Diag g , and the image of the moduli space under the period map has just one branch through W 1 g for each alkane. Each branch is smooth and has an explicit description as a vector bundle of rank g − 1 over a domain that contains W 1 g .
Highlights
The classical Schottky problem is that of describing the period locus Jg of period matrices of complex algebraic curves of genus g as a subvariety of Siegel space Hg .This problem naturally extends to higher dimensions; for example, an algebraic surface of positive geometric genus g has a period matrix that arises from integrating 2-forms around 2-cycles, and the Schottky problem becomes that of describing the image of the moduli space under the multi-valued period map
This picture is replicated for connected Jacobian elliptic surfaces, which form the simplest class of algebraic surfaces after K3 and abelian surfaces
In the period domain for such surfaces of geometric genus g there is a locus W1g that is analogous to Diagg, and the image of the moduli space under the period map has just one branch through W1g for each alkane
Summary
We are only concerned here with local aspects of the geometry of the situation, for which the fact that the period map is multi-valued is irrelevant We consider this problem for connected Jacobian elliptic surfaces of geometric genus g. There is a subdomain W1g of the period domain Vg for connected Jacobian elliptic surfaces which corresponds to trees of g special Kummer surfaces, whose definition is recalled below, and which is isomorphic to Hg1+1 We regard this as the analogue of the locus Diagg of diagonal matrices in Hg, which corresponds to trees of elliptic curves. I am grateful to Paolo Cascini, Bob Friedman, Mark Gross, Dave Morrison and Richard Taylor for some valuable conversation and correspondence and to Hershel Farkas, Sam Grushevsky and Riccardo Salvati Manni for their encouragement
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