Some loci in Teichmüller space for genus seven defined by vanishing thetanulls (Q1315080)

The author considers vanishing properties of the theta function for a Riemann surface of genus seven, which are equivalent according to the Riemann vanishing theorem, to the existence on \(W_ 7\) of four complete half-canonical linear series. The following theorem is proved: Suppose \(W_ 7\), Riemann surface of genus 7, admits 4 complete half- canonical linear series \(g^ 1_ 6\), \(h^ 1_ 6\), \(k^ 1_ 6\) and \(l^ 2_ 6\) whose sum is bicanonical. Then one of the following possibilities holds: (i) \(W_ 7\) is hyperelliptic, (ii) \(W_ 7\) is elliptic-hyperelliptic, (iii) \(W_ 7\) is in \({\mathcal N}_ 7\).