2026/03/03 by Fabrizio Catanese, Noah Ruhland
#math.AG #math.CV
We consider (1,1)-surfaces, namely, minimal compact complex surfaces S with pg (S) =KS2=1: for these the bicanonical map is a covering of degree 4 of the plane ℙ2. 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 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.