The unirational components of the strata of genus $11$ curves with several pencils of degree $6$ in ${\mathcal{M}}_{11}$

Abstract

We show that the strata ${\mathcal{M}}_{11,6}(k)\subset {\mathcal{M}}_{11,6}$ of $6$-gonal curves of genus $11$, equipped with $k$ mutually independent and type I pencils of degree six, have a unirational irreducible component for $5\le k\le 9$. The unirational families arise from degree $9$ plane curves with $4$ ordinary triple and $5$ ordinary double points that dominate an irreducible component of expected dimension. We will further show that the family of degree $8$ plane curves with $10$ ordinary double points covers an irreducible component of excess dimension in ${\mathcal{M}}_{11,6}(10)$.