Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves

Abstract

We consider $(1,1)$-surfaces, namely, minimal compact complex surfaces $S$ with $p_g (S) =K_S^2=1$: for these the bicanonical map is a covering of degree $4$ of the plane $\mathbb{P}^2$. And we answer a question posed by Meng Chen, whether they can contain a genus 2 pencil (this is the standard reason of failure of birationality of the bicanonical map). Our main theorem says that those which admit a genus 2 pencil form an irreducible subvariety of codimension $3$ in their moduli space $\frak M_{[1,1]}$; moreover, the general such surface admits exactly $12$ such pencils. The real fun is to relate this variety to the geometry of pencils of conics in the plane everywhere tangent to a cubic curve and a line. We investigate the corresponding variety $\mathcal{T}$ of triples and provide explicit equations using the classical theory of envelopes: among others, equations given in terms of the Weierstrass normal form of the cubic.