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Ciro Ciliberto

Publications and source records attributed to Ciro Ciliberto.

At least 55 records · Page 3Linked to original sources
arXiv2023

Irreducible unirational and uniruled components of moduli spaces of polarized Enriques surfaces

We give an explicit description of the irreducible components of the moduli spaces of polarized Enriques surfaces in terms of decompositions of the polarization as an effective sum of isotropic classes. We prove that infinitely many of these components are unirational (resp. uniruled). In particular, this applies to components of arbitrarily large genus $g$ and $ϕ$-invariant of the polarization.

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arXiv2022

Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds

Let $X$ be any smooth prime Fano threefold of degree $2g-2$ in $\mathbb{P}^{g+1}$, with $g \in \{3,\ldots,10,12\}$. We prove that for any integer $d$ satisfying $\left\lfloor \frac{g+3}{2} \right\rfloor \leq d \leq g+3$ the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree $d$ in $X$ is nonempty and has a component of dimension $d$, which is furthermore reduced except for the case when $(g,d)=(4,3)$ and $X$ is contained in a singular quadric. Consequently, we deduce that the moduli space of rank--two slope--stable $ACM$ bundles $\mathcal{F}_d$ on $X$ such that $\det(\mathcal{F}_d)=\mathcal{O}_X(1)$, $c_2(\mathcal{F}_d)\cdot \mathcal{O}_X(1)=d$ and $h^0(\mathcal{F}_d(-1))=0$ is nonempty and has a component of dimension $2d-g-2$, which is furthermore reduced except for the case when $(g,d)=(4,3)$ and $X$ is contained in a singular quadric. This completes the classification of rank-two $ACM$ bundles on prime Fano threefolds. Secondly, we prove that for every $h \in \mathbb{Z}^+$ the moduli space of stable Ulrich bundles $\mathcal{E}$ of rank $2h$ and determinant $\mathcal{O}_X(3h)$ on $X$ is nonempty and has a reduced component of dimension $h^2(g+3)+1$; this result is optimal in the sense that there are no other Ulrich bundles occurring on $X$. This in particular shows that any prime Fano threefold is Ulrich wild.

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arXiv2022

Two letters by Guido Castelnuovo

In this expository paper we transcribe two letters by Guido Castelnuovo, one to Francesco Severi, the other to Beniamino Segre, and explain the contents of both, which basically focus on the quest for an algebraic proof of the equality between the analytic and the arithmetic irregularity and of the closedness of regular 1-forms on a complex, projective, algebraic surface. Such an algebraic proof has been found only in the 1980's by Deligne and Illusie.

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arXiv2022

Unirationality of varieties described by families of projective hypersurfaces

Let $\mathscr{X}\to W$ be a flat family of generically irreducible hypersurfaces of degree $d\geq 2$ in $\PP^n$ with singular locus of dimension $t$, with $W$ unirational of dimension $r$. We prove that if $n$ is large enough with respect to $d$, $r$ and $t$, then $\mathscr{X}$ is unirational. This extends results in \cite {Pre, HMP}.

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arXiv2022

Double covers and extensions

In this paper we consider double covers of the projective space in relation with the problem of extensions of varieties, specifically of extensions of canonical curves to $K3$ surfaces and Fano 3-folds. In particular we consider $K3$ surfaces which are double covers of the plane branched over a general sextic: we prove that the general curve in the linear system pull back of plane curves of degree $k\geq 7$ lies on a unique $K3$ surface. If $k\leq 6$ the general such curve is instead extendable to a higher dimensional variety. In the cases $k=4,5,6$, this gives the existence of singular index $k$ Fano varieties of dimensions 8, 5, 3, and genera 17, 26, 37 respectively. For $k = 6$ we recover the Fano variety $\mathbf{P}(3, 1, 1, 1)$, one of only two Fano threefolds with canonical Gorenstein singularities with the maximal genus 37, found by Prokhorov. We show that the latter variety is no further extendable. For $k=4$ and $5$ these Fano varieties have been identified by Totaro. We also study the extensions of smooth degree 2 sections of $K3$ surfaces of genus 3. In all these cases, we compute the co-rank of the Gauss--Wahl maps of the curves under consideration. Finally we observe that linear systems on double covers of the projective plane provide superabundant logarithmic Severi varieties.

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arXiv2022

Ulrich bundles on a general blow--up of the plane

We prove that on $X_n$, the plane blown--up at $n$ general points, there are Ulrich line bundles with respect to a line bundle corresponding to curves of degree $m$ passing simply through the $n$ blown--up points, with $m\leq 2\sqrt{n}$ and such that the line bundle in question is very ample on $X_n$. We prove that the number of these Ulrich line bundles tends to infinity with $n$. We also prove the existence of slope--stable rank--$r$ Ulrich vector bundles on $X_n$, for $n\geq 2$ and any $r \geq 1$ and we compute the dimensions of their moduli spaces. These computations imply that $X_n$ is {Ulrich wild}.

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arXiv2022

Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

We prove the existence of various families of irreducible homaloidal hypersurfaces in projective space $\mathbb P^ r$, for all $r\geq 3$. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of dual hypersurfaces to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. These examples fit non--classical versions of de Jonquières transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan--Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision to the present knowledge.

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arXiv2022

Severi varieties on blow--ups of the symmetric square of an elliptic curve

We prove that certain Severi varieties of nodal curves of positive genus on general blow-ups of the twofold symmetric product of a general elliptic curve are non-empty and smooth of the expected dimension. This result, besides its intrinsic value, is an important preliminary step for the proof of nonemptiness of Severi varieties on general Enriques surfaces in arXiv:2109.10735.

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arXiv2021

Wahl maps and extensions of canonical curves and K3 surfaces

Let $C$ be a smooth projective curve of genus $g \geq 11$, non-tetragonal, considered in its canonical embedding in $\mathbf{P}^{g-1}$. We prove that $C$ is a linear section of an arithmetically Gorenstein normal variety $Y$ in $\mathbf{P}^{g+r}$, not a cone, with $\dim(Y)=r+2$ and $ω_Y=\mathcal{O}_Y(-r)$, if the Gauss--Wahl map of $C$ has corank larger or equal than $r+1$. This relies on previous work of Wahl and Arbarello-Bruno-Sernesi; a partial converse is given via a theorem of Lvovski. We derive a similar result for $K3$ surfaces: Let $(S,L)$ be a polarized $K3$ surface of genus $g \geq 11$, non-tetragonal, and considered in its embedding in $|L|^\vee \cong \mathbf{P}^g$. It is a linear section of a variety $Y$ as above if $H^1(T_S \otimes L^\vee)$ has dimension larger or equal than $r$. We give various applications, including one to the following forgetful modular map: Let $\mathcal{K}_g$ be the moduli space of polarized $K3$ surfaces of genus $g$, and $\mathcal{KC}_g$ the space of pairs $(S,C)$ with $C$ a smooth curve on $S$ and $(S,\mathcal{O}_S(C)) \in \mathcal{K}_g$; we consider the map $c_g: (S,C) \in \mathcal{K}_g \mapsto C \in \mathcal{M}_g$. If $g \geq 11$, we show that this map has smooth fibres over the locus of non-tetragonal curves, with fibre-dimension over non-tetragonal $C$ the corank of the Gauss--Wahl map of $C$ minus one.

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arXiv2021

Cones of lines having high contact with general hypersurfaces and applications

Given a smooth hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d\geqslant 2$, we study the cones $V^h_p\subset \mathbb{P}^{n+1}$ swept out by lines having contact order $h\geqslant 2$ at a point $p\in X$. In particular, we prove that if $X$ is general, then for any $p\in X$ and $2 \leqslant h\leqslant \min\{ n+1,d\}$, the cone $V^h_p$ has dimension exactly $n+2-h$. Moreover, when $X$ is a very general hypersurface of degree $d\geqslant 2n+2$, we describe the relation between the cones $V^h_p$ and the degree of irrationality of $k$--dimensional subvarieties of $X$ passing through a general point of $X$. As an application, we give some bounds on the least degree of irrationality of $k$--dimensional subvarieties of $X$ passing through a general point of $X$, and we prove that the connecting gonality of $X$ satisfies $d-\left\lfloor\frac{\sqrt{16n+25}-3}{2}\right\rfloor\leqslant\conngon(X)\leqslant d-\left\lfloor\frac{\sqrt{8n+1}+1}{2}\right\rfloor$.

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arXiv2020

On secant defective varieties, in particular of dimension 4

In this paper we prove some general results on secant defective varieties. Then we focus on the 4--dimensional case and we give the full classification of secant defective 4--folds. This paper has been inspired by classical work by G. Scorza,

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arXiv2020

On the infinitesimal Terracini lemma

In this paper we prove an infinitesimal version of the classical Terracini Lemma for 3--secant planes to a variety. Precisely we prove that if $X\subseteq \PP^r$ is an irreducible, non--degenerate, projective complex variety of dimension $n$ with $r\geq 3n+2$, such that the variety of osculating planes to curves in $X$ has the expected dimension $3n$ and for every $0$--dimensional, curvilinear scheme $γ$ of length 3 contained in $X$ the family of hyperplanes sections of $X$ which are singular along $γ$ has dimension larger that $r-3(n+1)$, then $X$ is $2$--secant defective.

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arXiv2020

Lines, conics, and all that

This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, etc.

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