arXiv subjects

C. G. Madonna

Publications and source records attributed to C. G. Madonna.

6 recordsLinked to original sources

arXiv2010

On some moduli spaces of bundles on K3 surfaces, II

We give many examples in which there exist infinitely many divisorial conditions on the moduli space of polarized K3 surfaces $(S,H)$ of degree $H^2=2g-2$, $g \geq 3$, and Picard number $rk N(S)=ρ(S)=2$ such that for a general K3 surface $S$ satisfying these conditions the moduli space of sheaves $M_S(r,H,s)$ is birationally equivalent to the Hilbert scheme $S[g-rs]$ of zero-dimensional subschemes of $S$ of lenght equal to $g-rs$. This result generalizes the main result of \cite{Nik1} when $g=rs+1$ and of \cite{Monat} when $r=s=2$, $g \geq 5$.

math.AG↗

arXiv2008

Curves and vector bundles on quartic threefolds

In this paper we study ACM vector bundles $\E$ of rank $k \geq 3$ on hypersurfaces $X_r \subset\Pj^4$ of degree $r \geq 1$. We consider here mainly the case of degree $r = 4$, which is the first unknown case in literature. Under some natural conditions for the bundle $\E$ we derive a list of possible Chern classes $(c_1,c_2,c_3)$ which may arise in the cases of rank $k=3$ and $k=4$, when $r=4$. For some cases among these we give the corresponding examples, the existence of all the other cases remaining under question.

math.AG↗

arXiv2006

On correspondences of a K3 surface with itself. IV

Let $X$ be a K3 surface with a polarization $H$ of the degree $H^2=2rs$, $r,s\ge 1$, and the isotropic Mukai vector $v=(r,H,s)$ is primitive. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(r,H,s)$ is again a K3 surface, $Y$. In \cite{Nik2} the second author gave necessary and sufficient conditions in terms of Picard lattice $N(X)$ of $X$ when $Y$ is isomorphic to $X$ (some important particular cases were also considered in math.AG/0206158, math.AG/0304415 and math.AG/0307355). Here we show that these conditions imply existence of an isomorphism between $Y$ and $X$ which is a composition of some universal geometric isomorphisms between moduli of sheaves over $X$, and geometric Tyurin's isomorphsim between moduli of sheaves over $X$ and $X$ itself. It follows that for a general K3 surface $X$ with $ρ(X)=\text{rk\}N(X)\le 2$ and $Y\cong X$, there exists an isomorphism $Y\cong X$ which is a composition of the geometric universal and the Tyurin's isomorphisms. This generalizes our recent results math.AG/0605362 and math.AG/0606239 to a general case.

math.AG↗

arXiv2006

On correspondences of a K3 surface with itself. III

Let $X$ be a K3 surface, and $H$ its primitive polarization of the degree $H^2=2rs$, $r,s\ge 1$. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(r,H,s)$ is again a K3 surface, $Y$. In math.AG/0206158, math.AG/0304415 and math.AG/0307355 (in general) we gave necessary and sufficient conditions in terms of Picard lattice $N(X)$ of $X$ when $Y$ is isomorphic to $X$, under the additional condition $H\cdot N(X)=\bz$. Here we show that these conditions imply existence of an isomorphism between $Y$ and $X$ which is a composition of some universal isomorphisms between moduli of sheaves over $X$, and Tyurin's isomorphsim between moduli of sheaves over $X$ and $X$ itself. It follows that for a general K3 surface $X$ with $H\cdot N(X)=\bz$ and $Y\cong X$, there exists an isomorphism $Y\cong X$ which is a composition of the universal and the Tyurin's isomorphisms. This generalizes our recent results math.AG/0605362 for $r=s=2$ on similar subject.

math.AG↗

arXiv2006

On a classical correspondence between K3 surfaces III

Let $X$ be a K3 surface, and $H$ its primitive polarization of the degree $H^2=8$. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(2,H,2)$ is again a K3 surface, $Y$. In math.AG/0206158 we gave necessary and sufficient conditions in terms of Picard lattice of $X$ when $Y$ is isomorphic to $X$. The proof of sufficient condition in math.AG/0206158, when $Y$ is isomorphic to $X$, used Global Torelli Theorem for K3 surfaces, and it was not effective. Here we give an effective variant of these results: its sufficient part gives an explicit isomorphism between $Y$ and $X$. We hope that our similar results in math.AG/0304415, math.AG/0307355, math.AG/0309348 for arbitrary primitive isotropic Mukai vector on a K3 surface also can be made effective.

math.AG↗

arXiv2001

ACM vector bundles on prime Fano threefolds and complete intersection Calabi Yau threefolds

In this paper we derive a list of all the possible indecomposable normalized rank--two vector bundles without intermediate cohomology on the prime Fano threefolds and on the complete intersection Calabi Yau threefolds, say $V$, of Picard number $ρ=1$. For any such bundle $\E$, if it exists, we find the projective invariants of the curves $C \subset V$ which are the zero-locus of general global sections of $\E$. In turn, a curve $C \subset V$ with such invariants is a section of a bundle $\E$ from our lists. This way we reduce the problem for existence of such bundles on $V$ to the problem for existence of curves with prescribed properties contained in $V$. In part of the cases in our lists the existence of such curves on the general $V$ is known, and we state the question about the existence on the general $V$ of any type of curves from the lists.

math.AG↗