Equations defining the periods of totally degenerate curves
A totally degenerate curve of genus \(g\) is a stable curve, all of whose components are rational and which has \(g\) additional ordinary double points. Mumford introduced for such a curve \(C\) a generalized Jacobian \(J(C)\), which is given by a totally degenerate symmetric \(g\times g\)- matrix, i.e. a matrix \((v_{ij})\) with \(v_{ii}=0\) and \(v_{ij} \neq 0\) for \(i \neq j\), the period matrix of \(C\). This gives a map \(\text{per}_ g\) from the space of totally degenerate curves of genus \(g\) to the torus of totally degenerate, symmetric \(g \times g\) matrices. The analogue of the classical Schottky problem consists in describing the image of \(\text{per}_ g\). In the paper an equation is derived which describes the locus of period matrices of totally degenerate curves of genus 4 whose intersection graph is simple. The result allows to deduce systems of equations for the periods of totally degenerate, irreducible curves of genus \(\geq 5\). Moreover, one can specialize the equation in genus 4 to obtain an equation describing the periods of totally degenerate hyperelliptic curves of genus 3.
scientific article; zbMATH DE number 203272
scientific article