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Ciro Ciliberto

Publications and source records attributed to Ciro Ciliberto.

At least 37 records · Page 2Linked to original sources
arXiv2025

Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps

In this paper, extending some ideas of Fano, we study the birational geometry of the Hilbert scheme of 0-dimensional subschemes of length 2 of a rational normal scroll. This fourfold has three elementary contractions associated to the three faces of its nef cone. We study natural projective realizations of these contractions. In particular, given a smooth rational normal scroll $S_{a,b}$ of degree $r$ in ${\mathbb P}^{r+1}$ with $1 \leq a \leq b$ and a+b=r, i.e., $S_{a,b}$ is the relative Proj of the vector bundle $O_{{\mathbb P}^1}(a)\oplus O_{{\mathbb P}^1}(b)$ embedded in ${\mathbb P}^{r+1}$ with its O(1) line bundle (from an abstract viewpoint $S_{a,b}\cong {\mathbb F}_{b-a}$), we consider the subvariety $X_{a,b}$ of the Grassmannian $G(1,r+1)$ described by all lines that are secant or tangent to $S_{a,b}$. The variety $X_{a,b}$ is the image of some of the aforementioned contractions, it is smooth if a>1, and it is singular at a unique point if a=1. We compute the degree of $X_{a,b}$ and the local structure of the singularity of $X_{a,b}$ when a=1. Finally we discuss in some detail the case r=4, originally considered by Fano, because the smooth hyperplane sections of $X_{2,2}$ and $X_{1,3}$ are the Fano 3-folds that appear as number 16 in the Mori-Mukai list of Fano 3-folds with Picard number 2. We prove that any smooth hyperplane section of $X_{2,2}$ is also a hyperplane section of $X_{1,3}$, and we discuss the GIT-stability of the smooth hyperplane sections of $X_{1,3}$ where $G$ is the subgroup of the projective automorphisms of $X_{1,3}$ coming from the ones of $S_{1,3}.$

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arXiv2024

Variations on Pascal's Theorem

In this paper we present a variety of statements that are in the spirit of the famous theorem of Pascal, often referred to as the Mystic Hexagon. We give explicit equations describing the conditions for $d+4$ points to lie on rational normal curves. A collection of problems of Pascal type are considered for quadric surfaces in $\bbP^3$. Finally we reprove, using computer algebra methods, a remarkable theorem of Richmond, Segre, and Brown, for quadrics in $\bbP^4$ containing five general lines.

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arXiv2024

On the degree of a modular map

Let $X$ be a general cubic hypersurface in $\mathbb P^4$. If $x\in X$ is a general point there are exactly six distinct lines in $X$ passing through $x$, that lie on the rank 3 quadric cone with vertex $x$ of lines that have intersection multiplicity at least 3 with $X$ in $x$. So there is a natural rational map $X\dasharrow \mathcal M_2$. In this paper we compute its degree to be 2074320.

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arXiv2024

Extensions of curves with high degree with respect to the genus

We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree $d$ such that $4g-4 \leq d \leq 4g+4$, where $g>1$ is the sectional genus (it is a classical result that for larger $d$ there are only cones). We apply this to the study of the extension theory of pluricanonical curves and genus $3$ curves, whenever they verify Property $N_2$, using and slightly expanding the theory of integration of ribbons of the authors and E.~Sernesi. We compute the corank of the relevant Gaussian maps, and we show that all ribbons over such curves are integrable, and thus there exists a universal extension. We carry out a similar program for linearly normal hyperelliptic curves of degree $d\geq 2g+3$. We classify surfaces having such a curve $C$ as a hyperplane section, compute the corank of the relevant Gaussian maps, and prove that all ribbons over $C$ are integrable if and only if $d=2g+3$. In the latter case we obtain the existence of a universal extension.

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arXiv2024

Enumeration of Terracini schemes

In this note we outline a way of computing the expected lenght of the Terracini scheme of a curve, when this scheme is expected to be finite and we give a closed formula for curves in $\p^4$. We also discuss the widely open case of varieties of higher dimension.

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arXiv2024

On the variety of tritangential planes to a general K3 surface of degree 6 and genus 4 in $\mP^4$

Let $S\subset \mP^4$ be a general K3 surface of degree 6 and genus 4. In this paper we study the irreducible variety $X_S$ of \emph{tritangential planes} to $S$ whose general point is a plane that intersects $S$ in a curvilinear scheme of length six supported at three non collinear points. The variety $X_S$ can be identified as the relevant part of the fixed locus of the so called \emph{Beauville involution} defined on the Hilbert scheme $S[3]$ of 0--dimensional schemes of length three of $S$. In this paper we prove that: (a) $X_S$ has dimension 3, is irreducible and smooth, except for 210 points that are at most of multiplicity 2 for $X_S$; (b) $X_S$, in its natural embedding in the Grassmannian $\mathbb G(2,4)\subset \mP^9$ of planes in $\mP^4$, has degree $152$.

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arXiv2024

Nonemptiness of Severi varieties on Enriques surfaces

Let $(S,L)$ be a general polarized Enriques surface, with $L$ not numerically 2-divisible. We prove the existence of regular components of all Severi varieties of irreducible $δ$-nodal curves in the linear system $|L|$, with $0\leq δ\leq p_a(L)-1$. This solves a classical open problem and gives a positive answer to a recent conjecture of Pandharipande--Schmitt, under the additional condition of non-2-divisibility.

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arXiv2023

On varieties whose general surface section has negative Kodaira dimension

In this paper, inspired by work of Fano, Morin and Campana--Flenner, we give a full projective classification of (however singular) varieties of dimension 3 whose general hyperplane sections have negative Kodaira dimension, and we partly extend such a classification to varieties of dimension $n\geq 4$ whose general surface sections have negative Kodaira dimension. In particular we prove that a variety of dimension $n\geq 3$ whose general surface sections have negative Kodaira dimension is birationally equivalent to the product of a general surface section times $\p^{n-2}$ unless (possibly) if the variety is a cubic hypersurface.

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arXiv2023

Quadric cones on the boundary of the Mori cone for very general blowups of the plane

We show the existence of cones over 8-dimensional rational spheres at the boundary of the Mori cone of the blow-up of the plane at $s\geq 13$ very general points. This gives evidence for De Fernex's strong $Δ$-conjecture, which is known to imply Nagata's conjecture. This also implies the existence of a multitude of good and wonderful rays as defined in Ciliberto-Harbourne-Miranda-Roé 2013.

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arXiv2023

Footnotes to a footnote of a paper by Fano

Following a suggestion in a footnote of the paper \cite {Fa} by Fano, we give a direct proof of the fact that one of the two Fano threefolds of maximal degree 72 in $\p^{38}$ is isomorphic to the anticanonically embedded weighted projective space $\p(1,1,4,6)$.

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arXiv2023

On the rationality of certain Fano threefolds

In this paper I study the rationality problem for Fano threefolds $X\subset \p^{p+1}$ of genus $p$, that are Gorenstein, with at most canonical singularities. The main results are: (1) a trigonal Fano threefold of genus $p$ is rational as soon as $p\geq 8$ (this result has already been obtained in \cite {PCS}, but we give here an independent proof); (2) a non--trigonal Fano threefold of genus $p\geq 7$ containing a plane is rational; (3) any Fano threefold of genus $p\geq 17$ is rational; (4) a Fano threefold of genus $p\geq 12$ containing an ordinary line $\ell$ in its smooth locus is rational.

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arXiv2023

A remark on the canonical degree of curves on smooth projective surface

The canonical degree $C.K_X$ of an integral curve on a smooth projective surface $X$ is conjecturally bounded from above by an expression of the form $A(g-1)+B$, where $g$ is the geometric genus of $C$ and $A$, $B$ are constants depending only on $X$. We prove that this conjecture holds with $A = -1$ under the assumptions $h^0(X, -K_X) = 0$ and $h^0(X, 2K_X + C) = 0$.

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arXiv2023

On deformations of the surfaces of bitangents to smooth quartic surfaces in $\mP^3$

We prove that the surface $S(X)$ of bitangent lines of a general smooth quartic surface $X$ in $\mP^3$ has unobstructed deformations of dimension $20=h^1(S(X), T_{S(X)})$. In addition, we show that the space of infinitesimal embedded deformations of $X$ injects into the one of $S(X)$. Finally we prove that there is a natural birational map from the 20--dimensional moduli space of (polarised) double coverings of EPW--sextics to the moduli space of regular surfaces $S$ with $p_g=45$ and $K_S^2=360$ polarised with a very ample line bundle $H$ such that $H^2=40$, $h^0(S, H)=6$: the map sends a double covering of a EPW--sextic in $\mP^5$ to the surface of double points of the EPW--sextic.

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arXiv2023

A remark on weighted Bounded Negativity for blow-ups of the projective plane

Motivated by the weighted Bounded Negativity Conjecture, we prove that all but finitely many reduced and irreducible curves $C$ on the blow-up of $\mathbb{P}^2$ at $n$ points satisfy the inequality $C^2 \ge \min \{-\frac{1}{12} n (C.L +27), -2 \}$, where $L$ is the pull-back of a line. This partially improves on some result by Laface and Pokora.

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arXiv2023

Ulrich bundles on Del Pezzo threefolds

We prove that for any $r \geq 2$ the moduli space of stable Ulrich bundles of rank $r$ and determinant $\mathcal O_X(r)$ on any smooth Fano threefold $X$ of index two is smooth of dimension $r^2+1$ and that the same holds true for even $r$ when the index is four, in which case no odd--rank Ulrich bundles exist. In particular this shows that any such threefold is Ulrich wild. As a preliminary result, we give necessary and sufficient conditions for the existence of Ulrich bundles on any smooth projective threefold in terms of the existence of a curve in the threefold enjoying special properties.

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arXiv2023

On the maximum number of complexes of a given degree containing subvarieties of Grassmannians

Let $\G(k,r)$ be the Grassmannian of $k$--subspaces in $\Proj^r$ embedded in $\Proj^{N(k,r)}$, with $N(k,r)={{r+1}\choose {k+1}}-1$, via the Plücker embedding. In this paper, extending some classical results by Gallarati (see \cite {Gal1,Gal2}), we give a sharp upper bound for the number of independent sections of $H^0(\G(k,r), Ø_{\G(k,r)}(m))$ vanishing on a subvariety $X$ of $\G(k,r)$ such that the union of the $k$--subspaces corresponding to the points of $X$ spans $\Proj^r$.

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arXiv2023

Remarks on the minimal genus of curves linearly moving on a surface

Given a smooth, irreducible, projective surface $S$, let $g(S)$ be the minimum geometric genus of an irreducible curve that moves in a linear system of positive dimension on $S$. We determine the value of this birational invariant for a general surface of degree $d$ in $\mathbb P^3$ and give a bound for $g(S)$ if $S$ is a general polarised K3 or abelian surface. As soon as this note appeared on the math arxiv, David Stapleton kindly pointed out to the author a paper by Ein and Lazarsfeld, in which in turn a paper by Konno was cited. Unfortunately the author was not aware of these two papers which contain results that strictly include the ones of this note. The author is very grateful to David Stapleton.

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