On surfaces with a canonical pencil

Details On surfaces with a canonical pencil On surfaces with a canonical pencil

2009-09-03

arxiv.org/abs/0909.0672v2

Mathematics

Algebraic Geometry

24 pages. v2: final version to appear on Mathematische Zeitschrift

Scientific paper

We classify the minimal surfaces of general type with $K^2 \leq 4\chi-8$ whose canonical map is composed with a pencil, up to a finite number of families. More precisely we prove that there is exactly one irreducible family for each value of $\chi \gg 0$, $4\chi-10 \leq K^2 \leq 4\chi-8$. All these surfaces are complete intersections in a toric $4-$fold and bidouble covers of Hirzebruch surfaces. The surfaces with $K^2=4\chi-8$ were previously constructed by Catanese as bidouble covers of $\PP^1 \times \PP^1$.

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