Geography of irreducible plane sextics

We complete the equisingular deformation classification of irreducible singular plane sextic curves. As a by-product, we also compute the fundamental groups of the complement of all but a few maximizing sextics

Algebraic cycles on a very special EPW sextic

Motivated by the Beauville-Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten-Bury et alii. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperk\"ahler fourfolds.Comment: 32 pages, to appear in Rend. Sem. Mat. Univ. Padova, feedback welcom

Periods of Double EPW-sextics

We study the indeterminacy locus of the period map for double EPW-sextics. We recall that double EPW-sextics are parametrized by lagrangian subspaces of the third wedge-product of a 6-dimensional complex vector-space. The indeterminacy locus is contained in the set of lagrangians containing a decomposable vector. The projectivization of the 3-dimensional support of such a decomposable vector contains a degeneracy subscheme which is either all of the plane or a sextic curve. We show that the period map is regular on any lagrangian A such that for all decomposables in A the corresponding degeneracy subscheme is a GIT-semistable sextic curve whose closure (in the semistable locus) does not contain a triple conic.Comment: We added a proof that the the period map of double EPW-sextics with isolated singularities is an open embedding into the complement of four explicit arithmetic divisors in the period space. Gave precise references to results of "Moduli of double EPW-sextics" (latest arXiv version) which will appear in Memoirs of the AM