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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.

##### MSC:
 14J50 Automorphisms of surfaces and higher-dimensional varieties 14J45 Fano varieties 14J70 Hypersurfaces and algebraic geometry
##### Keywords:
Fano surfaces; curves of genus 2
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