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Ciro Ciliberto

Publications and source records attributed to Ciro Ciliberto.

At least 91 records · Page 5Linked to original sources
arXiv2014

Limits of pluri-tangent planes to quartic surfaces

We describe, for various degenerations $S\to Δ$ of quartic $K3$ surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits as $t\in Δ^*$ tends to 0 of the Severi varieties $V_δ(S_t)$, parametrizing irreducible $δ$-nodal plane sections of $S_t$. We give applications of this to (i) the counting of plane nodal curves through base points in special position, (ii) the irreducibility of Severi varieties of a general quartic surface, and (iii) the monodromy of the universal family of rational curves on quartic $K3$ surfaces.

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arXiv2013

Abelian varieties in Brill--Noether loci

In this paper, improving on results of Abramovich- Harris and Debarre- Fahlaoui we give the full classification of curves C of genus g such that a Brill--Noether locus W^ s_d(C), strictly contained in the jacobian J(C) of C, contains a variety Z stable under translations by the elements of a positive dimensional abelian subvariety A contained in J(C) and such that dim(Z)=d-dim(A)-2s, i.e., the maximum possible for such a Z.

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arXiv2013

On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds

In this paper we study the gonality of the normalizations of curves in the linear system $|H|$ of a general primitively polarized complex $K3$ surface $(S,H)$ of genus $p$. We prove two main results. First we give a necessary condition on $p, g, r, d$ for the existence of a curve in $ |H|$ with geometric genus $g$ whose normalization has a $g^ r_d$. Secondly we prove that for all numerical cases compatible with the above necessary condition, there is a family of \emph{nodal} curves in $|H|$ of genus $g$ carrying a $g^1_k$ and of dimension equal to the \emph{expected dimension} $\min\{2(k-1),g\}$. Relations with the Mori cone of the hyperkähler manifold $\Hilb^ k(S)$ are discussed.

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arXiv2013

On Schoen surfaces

We give a new construction of the irregular, generalized Lagrangian, surfaces of general type with p_g=5, χ=2, K^2=8, recently discovered by Chad Schoen. Our approach proves that, if S is a general Schoen surface, its canonical map is a finite morphism of degree 2 onto a canonical surface with invariants p_g=5, χ=6, K^2=8, a complete intersection of a quadric and a quartic hypersurface in P^4, with 40 even nodes.

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arXiv2013

On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds (first version, superseded by arXiv:1204.4838)

In this paper we study the gonality of the normalizations of curves in the linear system $|H|$ of a general primitively polarized $K3$ surface $(S,H)$ of genus $p$. We prove two main results. First we give a necessary condition on $p, g, k$ for the existence of a curve in $|H|$ with geometric genus $g$ whose normalization has a $g^ 1_k$. Secondly we prove that for $p$ even and all numerical cases compatible with the above necessary condition, there is a family of \emph{nodal} curves $|H|$ with the given $g,k$ and of dimension equal to the \emph{expected dimension} $\min\{2(k-1),g\}$. For odd $p$ the result is only slightly less sharp. Relations with the Mori cone of the hyperkähler manifold $\Hilb^ k(S)$ and with conjectures by Hassett-Tschinkel and by Huybrechts-Sawon are discussed. This version is superseded by the new submission arXiv:1204.4838 where Theorem 0.1 is improved to include the missing case and the degeneration argument in its proof is made considerably simpler. Since the degeneration argument in the present version is of a different type, and may be useful for other purposes, we choose to keep this submission as well.

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arXiv2011

On the classification of OADP varieties

The main purpose of this paper is to show that OADP varieties stand at an important crossroad of various main streets in different disciplines like projective geometry, birational geometry and algebra. This is a good reason for studying and classifying them. Main specific results are: (a) the classification of all OADP surfaces (regardless to their smoothness); (b) the classification of a relevant class of normal OADP varieties of any dimension, which includes interesting examples like lagrangian grassmannians. Following [PR], the equivalence of the classification in (b) with the one of quadro-quadric Cremona transformations and of complex, unitary, cubic Jordan algebras are explained.

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arXiv2011

Scrolls and hyperbolicity

Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in P3 of degree d >=5. We show, along the same lines, boundedness of families of curves of small enough genera on general surfaces in P3. We also show that there exist Kobayashi hyperbolic surfaces in P3 of degree d = 7 (a result so far unknown), and give a new construction of such surfaces of degree d = 6. Finally we provide some new lower bounds for geometric genera of surfaces lying on general hypersurfaces of degree 3d > 15 in P4.

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arXiv2010

On a theorem of Castelnuovo and applications to moduli

In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. Then we classify linear systems whose dimension belongs to certain intervals which naturally arise from Castelnuovo's theorem. Finally we make an application to the following moduli problem: what is the maximum number of moduli of curves of geometric genus $g$ varying in a linear system on a surface? It turns out that, for $g\ge 22$, the answer is $2g+1$, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.

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arXiv2010

Plucker-Clebsch formula in higher dimension

Let $S\subset\Ps^r$ ($r\geq 5$) be a nondegenerate, irreducible, smooth, complex, projective surface of degree $d$. Let $δ_S$ be the number of double points of a general projection of $S$ to $\Ps^4$. In the present paper we prove that $ δ_S\leq{\binom {d-2} {2}}$, with equality if and only if $S$ is a rational scroll. Extensions to higher dimensions are discussed.

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arXiv2010

The rank of the 2nd Gaussian map for general curves

We prove that, for the general curve of genus g, the 2nd Gaussian map is injective if g <= 17 and surjective if g >= 18. The proof relies on the study of the limit of the 2nd Gaussian map when the general curve of genus g degenerates to a general stable binary curve, i.e. the union of two rational curves meeting at g+1 points.

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arXiv2009

Birational classification of curves on rational surfaces

In this paper we consider the birational classification of pairs (S,L), with S a rational surfaces and L a linear system on S. We give a classification theorem for such pairs and we determine, for each irreducible plane curve B, its "Cremona minimal" models, i.e. those plane curves which are equivalent to B via a Cremona transformation, and have minimal degree under this condition.

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arXiv2008

On the Dimension of Secant Varieties

We generalize Zak's theorems on tangencies and on linear normality as well as Zak's definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained.

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arXiv2008

Homogeneous interpolation on ten points

In this paper we prove that for all pairs $(d,m)$ with $d/m \geq 174/55$, the linear system of plane curves of degree $d$ with ten general base points of multiplicity $m$ has the expected dimension.

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