ALGEBRAIC TRANSFORMATION GROUPs SEMINAR

Abstract: By a theorem of Catanese, the canonical model of a minimal surface of general type whose canonical class has self-intersection one and whose geometric genus is one is a complete intersection of two sextics in P(1,2,2,3,3). We describe the automorphism group of these surfaces and prove that its order is at most 72. We also classify the surfaces attaining the bound: two surfaces with singular canonical model and a one-parameter family of smooth surfaces. For this smooth family we compute a basis of the primitive cohomology given by residues of monomials, together with the intersection form. We then show that the Torelli principle for automorphisms fails: the primitive part of the Néron–Severi lattice contains a root system of rank twelve whose reflections are Hodge isometries. As a result, the automorphism group has index at least 55.296 in the group of Hodge isometries of the primitive lattice taken up to sign. This is joint work with Jorge Duque Franco and Alvaro Liendo.