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Ciro Ciliberto

Publications and source records attributed to Ciro Ciliberto.

At least 19 recordsLinked to original sources
arXiv2026

Components of simple and non--simple type of Hurwitz schemes

Let $\mathcal{H}_{g \to b,d; \mathbf{e}}$, with $\mathbf{e}=(e_1,\ldots, e_n)$, be the Hurwitz space, parametrizing all morphisms $\pi: C\to B$ of degree $d$, with $n$ points $x_1,\ldots, x_n\in C$ of ramification order $e_1,\ldots, e_n$ respectively, and where $C$ and $B$ are smooth, irreducible, projective curves of genera $g$ and $b$ respectively. In this paper we study the question of when there exist components of $\mathcal{H}_{g \to b,d; \mathbf{e}}$ whose members $\pi: C \to B$ all factor through an intermediate curve, in which case we say that these components are \emph{of non--simple type}. We give necessary and sufficient conditions for the existence of components of non--simple type. Then we prove that for $b\geq 2$ there are always components of simple type, and for $b\in \{0,1\}$ there are such components under suitable sufficient conditions. However there are easy examples for $b\in \{0,1\}$ in which there are never components of simple type.

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arXiv2026

Contact Invariants for Plane Curves in a Pencil

Let $\calP$ be a general pencil of curves of degree $d$ in the projective plane. In this paper we review the computation of the number of curves in $\calP$ that have a hyperflex line, a flex bitangent line or a tritangent line. Then we focus on the curves in the dual plane described by the flex tangents and the bitangents of the curves of $\calP$ and the curves in the original plane described by the flexes and the points of bitangencies of the curves in $\calP$. Some of these curves have been studied already: we mainly focus here on the ones that still have not been treated systematically, and we compute their degree, genus, and singularities.

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arXiv2026

A remark on isolated complex hypersurface singularities

This is now an expository note about the following classical problem. Let $(X, \bf 0)$ be the germ of a hypersurface in $(\mathbb C^n,\bf 0)$ with an ordinary singularity of multiplicity $m$ at the origin $\bf 0$. A natural question to ask is whether $X$ and its tangent cone at the origin are analytically isomorphic. The answer is negative in general, in view of a theorem of Kioji Saito. However there is an integer $D(n,m)>m$ such that, given a \emph{regular} homogeneous polynomial $f(x_1,\ldots, x_n)$ of degree $m$ (this means that $\{ f=0\}$ is a smooth hypersurface in $\PP^{n-1}$) then, for all $d\geq D(n,m)$, any convergent power series of the form $g=f+ o(d)$ (here, as usual, $o(d)$ stays for a power series of order at least $d$), defines a germ $\{ g=0\}$ which is analytically equivalent to the germ $\{ f=0\}$. In this note we compute $D(n,m)$ explicitly as $n(m-2)+1$. We also give an extension to the case in which $f$ is a quasihomogeneous polynomial. It was pointed out that the value of $D(n,m)$ was already known by \cite[Exercise 7.31]{D}.

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arXiv2026

Constructions for rational multiple planes

A finite, normal cover $f: X\longrightarrow \bbP^2$ of degree $m\geq 3$ (the case $m=2$ is well known and we do not consider it in this paper) is called \emph{simple}, if there is a pencil $\mathcal P$ of rational curves of $\bbP^2$ such that the pull back via $f$ of $\mathcal P$ is a pencil of rational curves on $X$. Up to Cremona equivalence $\mathcal P$ can be assumed to be the pencil of lines through a fixed point $p\in \bbP^2$. If $\frakB$ is the branch curve of such a multiple plane, the general line through $p$ has to intersect $\frakB$ in $2m-2$ branch points (counted with multiplicities). If $p$ is not one of these branch points, then the multiple plane is said to be \ \emph{simpler}. \ In that case the branch curve will have a point of multiplicity $\deg(\frakB)-2m+2$ at $p$. In this paper we classify, under suitable generality conditions for the branch curve, { simpler } triple planes up to Cremona equivalence (they belong to infinitely many non--Cremona equivalent families) and we give examples of infinitely many non--Cremona equivalent families of {simpler } multiple planes of degree $m\geq 4$.

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arXiv2026

Rational elliptic surfaces with six singular double fibres

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible curve of genus one and $f$ has a section. In this paper, we classify rational elliptic surfaces with section that have exactly six singular fibres, each counted with multiplicity two. The fibres that appear with multiplicity exactly two are either of type $II$ or of type $I_2$ of the Kodaira classification. We interpret our classification from various viewpoints: a pencil of plane cubic curves, the Weierstrass equation, a double cover of $\bbF_2$ branched over an appropriate trisection of the ruling of $\bbF_2$ plus the negative section, a double cover of the plane branched along a quartic curve, plus the datum of a point on the plane. Moreover, either we give explicit normal forms for the plane quartic curve, or we indicate how to find it.

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arXiv2026

On nodal deformations of singular surfaces in $\mathbb P^3$

In this paper we study nodal deformations of singular surfaces $S\subset \mathbb P^3$. In particular we consider the case in which $S$ has an isolated singularity of multiplicity $m$ and the case in which $S$ has only ordinary singularities along a line.

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arXiv2025

Boundedness Results for Planar Linear Systems Assuming The Segre-Harbourne-Gimigliano-Hirschowitz Conjecture

Let $X_n$ be the projective plane blown up at $n \geq 10$ general points. In this paper we give several consequences of the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture, that pertain to complete linear systems on $X_n$. We begin by classifying such systems $|C|$ with general irreducible member of genus $g \geq 2$ (up to Cremona equivalence), in terms of invariants of the adjoint systems $|C+mK|$. We then use this to prove that, for fixed $n \geq 10$ and $g\geq 2$, up to the action of the Cremona group, there exist finitely many complete linear systems on $X_n$ whose general member is irreducible of genus $g$. Further, there is a function $g\mapsto n(g)$ such that every such (effective) system is Cremona equivalent to a system in $X_{n(g)}$. The latter result is based on the explicit computation of the minimum possible self-intersection of an irreducible linear system with given $n$ and $\dim(|C|)$. We classify those systems which achieve the minimal self-intersection. We also classify the systems with $C^2 \leq 5$, whether or not they have minimal $C^2$ for the given $n$ and dimension. We finish by proving several statements concerning systems that are base-point-free, and systems that give birational maps to their image.

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arXiv2025

The Caporaso-Harris-Ran degeneration principle: proof and applications

Severi varieties are the parameter spaces for curves with prescribed homology class and genus on a smooth surface. We describe their limits along degenerations of surfaces, with a view towards the enumeration of curves. This includes a complete proof of the Caporaso-Harris recursive formula, with all the necessary background on deformations of curves and singularities.

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arXiv2025

On some boundary divisors in the moduli spaces of stable Horikawa surfaces with $K^2=2p_g-3$

We describe the normal stable surfaces with K^2=2p_g-3 and p_g>14 whose only non canonical singularity is a cyclic quotient singularity of type 1/4k(1,2k-1) and the corresponding locus D inside the KSBA moduli space of stable surfaces. More precisely, we show that: (1) a general point of any irreducible component of D corresponds to a surface with a singularity of type 1/4(1,1), (2) the closure of D is a divisor contained in the closure of the Gieseker moduli space of canonical models of surfaces with K^2=2p_g-3 and intersects all the components of such closure, and (3) the KSBA moduli space is smooth at a general point of D. In addition, we show that D has 1 or 2 irreducible components, depending on the residue class of p_g modulo 4.

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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

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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