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Genus 2 curves configurations on Fano surfaces. (English) Zbl 1215.14047
Let \(S\) be the Fano surface which parameterizes lines of a cubic threefold \(F\). The author considers the structure of genus \(2\) curves on \(S\). These curves provide an obstruction for the very-ampleness of the cotangent map to \(\mathbb P (H^0\Omega_S^*)\). The author shows that genus \(2\) curves in \(S\) are connected with involutions of certain type II of \(S\). So, the author takes the subgroup \(\Gamma\) of the automorphisms group \(\text{Aut}(S)\), generated by involutions of type II, and starts a classification of Fano surfaces in terms of \(\Gamma\). For instance, the author constructs and studies Fano surfaces for which \(\text{Aut}(S)\) contains a subgroup \(G\), formed by involutions of type II, which is either \(\mathbb Z/2\mathbb Z\) or the dihedral group \(\mathbb D_n\), \(n=2,3,4,5\), or the alternating group \(\mathbb A_5\) or \(\text{PSL}_2(\mathbb F_{11})\). In these cases, the structure of the corresponding curves of genus \(2\) on \(S\), and their degenerations, is described.

14J50 Automorphisms of surfaces and higher-dimensional varieties
14J45 Fano varieties
14J70 Hypersurfaces and algebraic geometry
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