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Carlo Madonna

Publications and source records attributed to Carlo Madonna.

5 recordsLinked to original sources
arXiv2014

EPW sextics and Hilbert squares of K3 surfaces

We prove that the Hilbert square $S^{[2]}$ of a very general primitively polarized K3 surface S of degree $d(n) = 2(4n^2 + 8n + 5)$, $n \geq 1$ is birational to a double Eisenbud-Popescu-Walter sextic. Our result implies a positive answers, in the case when $r$ is even, to a conjecture of O'Grady: On the Hilbert square of a very general K3 surface of genus $r^2 + 2$, $r \geq 1$ there is an antisymplectic involution. We explicitly give this involution on $S^{[2]}$ in term of the corresponding EPW polarization on it.

math.AG↗
arXiv2003

On a classical correspondence between K3 surfaces II

Let X be a K3 surface and H a primitive polarization of degree H^2=2a^2, a>1. The moduli space of sheaves over X with the isotropic Mukai vector (a,H,a) is again a K3 surface Y which is endowed by a natural nef element h with h^2=2. We give necessary and sufficient conditions in terms of Picard lattices N(X) and N(Y) when Y\cong X, generalising our results math.AG/0206158 for a=2. E.g. we show that Y\cong X if for one of α=\pm 1,\pm 2 which is coprime to a there exists h_1\in N(X) such that h_1^2= 2αa, H\cdot h_1\equiv 0\mod αa, and the primitive sublattice [H,h_1]_{pr} \subset N(X) contains x such that $x\cdot H=1$. We find all divisorial conditions on moduli of (X,H) (i.e for Picard number 2) which imply Y\cong X and H\cdot N(X)=Z. Some of these conditions were found in different form by A.N. Tyurin in 1987.

math.AG↗
arXiv2002

On a classical correspondence between K3 surfaces

Let X be a K3 surface which is intersection of three (a net P^2) of quadrics in P^5. The curve of degenerate quadrics has degree 6 and defines a double covering of P^2 K3 surface Y ramified in this curve. This is a classical example of a correspondence between K3 surfaces which is related with moduli of vector bundles on K3 studied by Mukai. When general (for fixed Picard lattices) X and Y are isomorphic? We give necessary and sufficient conditions in terms of Picard lattices of X and Y. E.g. for Picard number 2, the Picard lattice of X and Y is defined by its determinant (-d) where d>0, d\equiv 1 \mod 8, and one of equations a^2-db^2=8 or a^2-db^2=-8 should have an integral solution (a,b). The set of these d is infinite: d\in {(a^2\mp 8)/b^2} where a and b are odd integers. This describes all possible divisorial conditions on 19- dimensional moduli of intersections of three quadrics in P^5 when Y\cong X. One of them when X has a line is classical, and corresponds to d=17. Similar considerations can be applied for a realization of an isomorphism (T(X)\otimes Q, H^{2,0}(X)) \cong (T(Y)\otimes Q, H^{2,0}(Y)) of transcendental periods over Q of two K3 surfaces X and Y by a fixed sequence of types of Mukai vectors.

math.AG↗