New surfaces with canonical map of high degree

Abstract

We give an algorithm that, for a given value of the geometric genus p g , computes all regular product-quotient surfaces with abelian group that have at most canonical singularities and have canonical system with at most isolated base points. We use it to show that there are exactly two families of such surfaces with canonical map of degree 32. We also construct a surface with q = 1 and canonical map of degree 24. These are regular surfaces with p g = 3 and base point free canonical system. We discuss the case of regular surfaces with p g = 4 and base point free canonical system.

A new infinite family of irregular algebraic surfaces with canonical map of degree 8

“…For details about the building data of abelian covers and their notations, we refer the reader to Section 1 and Section 2 of R. Pardini's work ([9]). For the sake of completeness, we recall some facts on Z 3 2 -covers, in a form which is convenient for our later constructions.…”

“…It is worth pointing out that C. Gleissner, R. Pignatelli and C. Rito constructed a family of surfaces with K 2 X = 24, p g (X) = 3, q (X) = 1 and d = 24 ([3]). Their example has a very similar construction as Z 3 2 -cover of P 1 × C branched on "fibers" of the obvious trivial fibrations.…”

“…These examples were constructed as double covers of the product surface of a non-hyperelliptic curve of genus 3 and the rational curve P 1 . In [8], some unlimited families of surfaces with canonical map of degree 8 were found by as Z 3 2 -covers of the first Hirzebruch surface F 1 or its blow-up in a point. In these all known examples, the image of the canonical map is a surface of minimal degree.…”

“…These surfaces are constructed as Z 3 2 -covers of the product surface of a smooth elliptic curve C and the rational curve P 1 . The building data {L χ , D σ } χ,σ of Z 3 2 -covers (see Section 2) are chosen such that there is a character χ ′ of Z 3 2 with arbitrarily large h 0 (L χ ′ + K P 1 ×C ) and that h 0 (L χ + K P 1 ×C ) vanishes for all other characters χ of Z 3 2 . From the decomposition of the space of 2-forms of the surfaces (see Proposition 2)…”

A surface with canonical map of degree 24

“…Only recently examples with d > 16 have been given, see [GPR18], [Rit17] for d = 24 and [GPR18] for d = 32. It follows from Beauville's proof that the limit cases d = 36, q = 0 and d = 27, q > 0 can only occur for surfaces with invariants p g = 3, q = 0, K 2 = 36 and p g = 3, q = 1, K 2 = 27,…”