# An effective construction of degree 8 surfaces in $P^4$ with 40 nodes

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Xavier Roulleau

created by risa on 16 Oct 2015

23 oct 2015
-- 16:45

Aula Dal Passo, Dip. Matematica, Università "Tor Vergata", Roma

**Abstract.**

Joint work with Joint work with Carlos Rito and Alessandra Sarti. Using smoothing technics, Schoen constructed a family of surfaces S with $K^2=2$, $c_2=16$, $q=4$, which have many remarkable properties. In a previous work with C. Ciliberto and M. Mendes-Lopes, we proved that a Schoen surface S are double cover of a degree 8 complete intersection surface in $P^4$ with 40 nodes (which is the maximal number possible). In this talk we construct in an effective way a degree 8 complete intersection surface Z in $P^4$ with 40 nodes, using the geometries of quartic K3 surfaces with 15 nodes, the Segre cubic threefold and the Igusa quartic threefold. A double cover S of Z branched over the 40 nodes has the same invariants as the Schoen surface S and we prove that it is not isogeneous to a higher quotient. We then exhibit an example of a surface Swith maximal Picard number and a large group of symmetries.