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Francesco Bastianelli

Publications and source records attributed to Francesco Bastianelli.

At least 19 recordsLinked to original sources
arXiv2026

Subvarieties of complete intersections of large degree

We study subvarieties of very general complete intersections $X\subset \mathbb{P}^n$ of multidegree $(d_1,\dots,d_c)$, when $d:= d_1+\dots +d_c$ is sufficiently large. In a seminal paper Ein proved that if $d\geq 2n-c-k+2$, any $k$-dimensional subvariety of $X$ is of general type and has positive geometric genus. We strengthen this result by obtaining the optimal bound $d\geq 2n-c-k$, provided that $n> 2c+k$. As a consequence, we characterize algebraic hyperbolicity of very general complete intersections $X\subset \mathbb{P}^n$ of codimension $c\leq \frac{n-3}{2}$. For lower values of $d$, we prove that if $\frac{3n-c+2}{2}\leq d\leq 2n-c-2$ and $(d_1,\dots,d_c)$ satisfies an additional numerical condition, then the only curves in $X$ that are not of general type are lines. Moreover, we describe the locus where positive dimensional orbits of points under rational equivalence must lie. We obtain our results by proving that, under suitable numerical conditions, subvarieties of $X$ that are not of general type must lie in the locus of $X$ covered by lines. The proof of this result relies on a generalization of the approach and techniques developed for hypersurfaces by Voisin, Clemens-Ran and the second author, combined with a Grassmannian technique introduced by Riedl-Yang.

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arXiv2025

Indecomposability of derived categories in families

Using the moduli space of semiorthogonal decompositions in a smooth projective family, introduced by the second, the third and the fourth author, we propose a novel approach to indecomposability questions for derived categories. Modulo a natural conjecture on the structure of the moduli space, we give both general results, and discuss interesting explicit examples of the behaviour of indecomposability in families, by relating it to the behaviour of the canonical base locus in families. These examples are symmetric powers of curves, certain regular surfaces of general type with large canonical base locus, and Hilbert schemes of points on surfaces. Indecomposability for symmetric powers of curves has been settled via other means, the other cases remain open and we expect that our analysis of the base locus will prove instrumental in finding unconditional proofs.

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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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arXiv2024

Moving curves of least gonality on symmetric products of curves

This paper is a sequel of arXiv:2208.00990. Let $C$ be a smooth complex projective curve of genus $g$ and let $C^{(k)}$ be its $k$-fold symmetric product. The covering gonality of $C^{(k)}$ is the least gonality of an irreducible curve $E\subset C^{(k)}$ passing through a general point of $C^{(k)}$. It follows from previous works of the authors that if $2\leq k\leq 4$ and $g\geq k+4$, the covering gonality of $C^{(k)}$ equals the gonality of $C$. In this paper, we prove that under mild assumptions of generality on $C$, the only curves $E\subset C^{(k)}$ computing the covering gonality of $C^{(k)}$ are copies of $C$ of the form $C+p$, for some point $p\in C^{(k-1)}$. As a byproduct, we deduce that the connecting gonality of $C^{(k)}$ (i.e. the least gonality of an irreducible curve $E\subset C^{(k)}$ connecting two general points of $C^{(k)}$) is strictly larger than the covering gonality.

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arXiv2023

Covering gonality of symmetric products of curves and Cayley-Bacharach condition on Grassmannians

Given an irreducible projective variety $X$, the covering gonality of $X$ is the least gonality of an irreducible curve $E\subset X$ passing through a general point of $X$. In this paper we study the covering gonality of the $k$-fold symmetric product $C^{(k)}$ of a smooth complex projective curve $C$ of genus $g\geq k+1$. It follows from a previous work of the first author that the covering gonality of the second symmetric product of $C$ equals the gonality of $C$. Using a similar approach, we prove the same for the $3$-fold and the $4$-fold symmetric product of $C$. A crucial point in the proof is the study of Cayley-Bacharach condition on Grassmannians. In particular, we describe the geometry of linear subspaces of $\mathbb{P}^n$ satisfying this condition and we prove a result bounding the dimension of their linear span.

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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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arXiv2019

Gonality of curves on general hypersurfaces

This paper concerns the existence of curves with low gonality on smooth hypersurfaces of sufficiently large degree. It has been recently proved that if $X\subset \mathbb{P}^{n+1}$ is a hypersurface of degree $d\geq n+2$, and if $C\subset X$ is an irreducible curve passing through a general point of $X$, then its gonality verifies $\mathrm{gon}(C)\geq d-n$, and equality is attained on some special hypersurfaces. We prove that if $X\subset \mathbb{P}^{n+1}$ is a very general hypersurface of degree $d\geq 2n+2$, the least gonality of an irreducible curve $C\subset X$ passing through a general point of $X$ is $\mathrm{gon}(C)=d-\left\lfloor\frac{\sqrt{16n+1}-1}{2}\right\rfloor$, apart from a series of possible exceptions, where $\mathrm{gon}(C)$ may drop by one.

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arXiv2019

Effective cycles on the symmetric product of a curve, I: the diagonal cone. (with an appendix by Ben Moonen)

In this paper and in its sequel [BKLV], we investigate the cone ${\rm Pseff}_n(C_d)$ of pseudoeffective $n$-cycles in the symmetric product $C_d$ of a smooth curve $C$. In the present paper, we study the convex-geometric properties of the cone generated by the $n$-dimensional diagonal cycles, which we call the $n$-dimensional diagonal cone. We prove that the $n$-dimensional diagonal cone is a perfect face of ${\rm Pseff}_n(C_d)$ along which ${\rm Pseff}_n(C_d)$ is locally finitely generated.

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arXiv2018

On complete intersections containing a linear subspace

Consider the Fano scheme $F_k(Y)$ parameterizing $k$-dimensional linear subspaces contained in a complete intersection $Y \subset \mathbb{P}^m$ of multi-degree $\underline{d} = (d_1, \ldots, d_s)$. It is known that, if $t := \sum_{i=1}^s \binom{d_i +k}{k}-(k+1) (m-k)\leqslant 0$ and $Π_{i=1}^sd_i >2$, for $Y$ a general complete intersection as above, then $F_k(Y)$ has dimension $-t$. In this paper we consider the case $t> 0$. Then the locus $W_{\underline{d},k}$ of all complete intersections as above containing a $k$-dimensional linear subspace is irreducible and turns out to have codimension $t$ in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general $[Y]\in W_{\underline{d},k}$ the scheme $F_k(Y)$ is zero-dimensional of length one. This implies that $W_{\underline{d},k}$ is rational.

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arXiv2018

Effective cycles on the symmetric product of a curve, II: the Abel-Jacobi faces

In this paper, which is a sequel of [BKLV], we study the convex-geometric properties of the cone of pseudoeffective $n$-cycles in the symmetric product $C_d$ of a smooth curve $C$. We introduce and study the Abel-Jacobi faces, related to the contractibility properties of the Abel-Jacobi morphism and to classical Brill-Noether varieties. We investigate when Abel-Jacobi faces are non-trivial, and we prove that for $d$ sufficiently large (with respect to the genus of $C$) they form a maximal chain of perfect faces of the tautological pseudoeffective cone (which coincides with the pseudoeffective cone if $C$ is a very general curve).

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arXiv2017

On irrationality of surfaces in $\mathbb{P}^3$

The degree of irrationality $irr(X)$ of a $n$-dimensional complex projective variety $X$ is the least degree of a dominant rational map $X\dashrightarrow \mathbb{P}^n$. It is a well-known fact that given a product $X\times \mathbb{P}^m$ or a $n$-dimensional variety $Y$ dominating $X$, their degrees of irrationality may be smaller than the degree of irrationality of $X$. In this paper, we focus on smooth surfaces $S\subset\mathbb{P}^3$ of degree $d\geq 5$, and we prove that $irr(S\times\mathbb{P}^{m})=irr(S)$ for any positive integer $m$, whereas $irr(Y)<irr(S)$ occurs for some $Y$ dominating $S$ if and only if $S$ contains a rational curve.

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arXiv2016

Gonality of curves on fundamental loci of first order congruences (Appendix to article of Ein, Lazarsfeld and Ullery)

This note is an appendix to 'Measures of irrationality for hypersurfaces of large degree' by L. Ein, R. Lazarsfeld and B. Ullery. We prove an existence result for families of curves having low gonality, and lying on fundamental loci of first order congruences of lines in $\mathbb{P}^{n+1}$. As an application, we follow the ideas of the main paper, and we present a slight refinement of a theorem included in it. In particular, we show that given a very general hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d \geq 3n-2 \geq 7$, and a dominant rational map $f \colon X \dashrightarrow \mathbb{P}^{n}$, then $deg(f) \geq d-1$, and equality holds if and only if $f$ is the projection from a point of $X$.

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arXiv2016

Measures of irrationality for hypersurfaces of large degree

We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if X is a very general smooth hypersurface of dimension n and degree d \ge 2n+1, then any dominant rational mapping from X to projective n-space must have degree at least d-1. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti-Eisenbud.

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arXiv2015

Subcanonical points on projective curves and triply periodic minimal surfaces in the Euclidean space

A point $p$ on a smooth complex projective curve $C$ of genus $g>3$ is subcanonical if the divisor $(2g-2)p$ is canonical. In the moduli space of pointed curves, the subcanonical locus is described by pairs $(C,p)$ as above, and it consists of three irreducible components of dimension $2g-1$. Apart from the hyperelliptic component $\mathcal{G}_g^{hyp}$, the other components $\mathcal{G}_g^{odd}$ and $\mathcal{G}_g^{even}$ depend on the parity of $h^0(C,(g-1)p)$, and their general points satisfy $h^0(C,(g-1)p)=1$ and $2$, respectively. In this paper, we study the subloci of pairs $(C,p)$ such that $h^0(C,(g-1)p)$ is at least $r+1$ and it has the same parity as $r+1$. In particular, we provide a lower bound on their dimension, and we prove its sharpness for $r<4$. As an application, we further give an existence result for triply periodic minimal surfaces immersed in the 3-dimensional Euclidean space, completing a previous result of the second author.

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arXiv2015

On large theta-characteristics with prescribed vanishing

Let $C$ be a smooth projective curve of genus $g\geq 2$. Fix an integer $r\geq 0$, and let $\underline{k}=(k_1,\ldots,k_n)$ be a sequence of positive integers with $k_1+\ldots+k_n=g-1$. We study $n$-pointed curves $(C,p_1,\ldots,p_n)$ such that the line bundle $L:=O_C\left(\sum_{i=1}^n k_i p_i\right)$ is a theta-characteristic such that $h^0\left(C,L\right)$ is at least $r+1$ and it has the same parity as $r+1$. We prove that they describe a sublocus $\mathcal{G}^r_g(\underline{k})$ of $\mathcal{M}_{g,n}$ having codimension at most $g-1+\frac{r(r-1)}{2}$. Moreover, for any $r\geq 0$, $\underline{k}$ as above, and $g$ greater than an explicit integer $g(r)$ depending on $r$, we present irreducible components of $\mathcal{G}^r_g(\underline{k})$ attaining the maximal codimension in $\mathcal{M}_{g,n}$, so that the bound turns out to be sharp.

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arXiv2011

The gonality theorem of Noether for hypersurfaces

It is well known since Noether that the gonality of a smooth plane curve of degree d>3 is d-1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is the minimum degree of a dominant rational map from X to $\mathbb{P}^k$. In this paper we are aimed at extending the assertion on plane curves to smooth hypersurfaces in $\mathbb{P}^n$ in terms of degree of irrationality. We prove that both surfaces in $\mathbb{P}^3$ and threefolds in $\mathbb{P}^4$ of sufficiently large degree d have degree of irrationality d-1, except for finitely many cases we classify, whose degree of irrationality is d-2. To this aim we use Mumford's technique of induced differentials and we shift the problem to study first order congruences of lines of $\mathbb{P}^n$. In particular, we also slightly improve the description of such congruences in $\mathbb{P}^4$ and we provide a bound on degree of irrationality of hypersurfaces of arbitrary dimension.

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arXiv2010

Galois closure and Lagrangian varieties

We use Galois closures of finite rational maps between complex projective varieties to introduce a new method for producing varieties such that the holomorphic part of the cup product map has non-trivial kernel. We then apply our result to the two-dimensional case and we construct a new family of surfaces which are Lagrangian in their Albanese variety. Moreover, we analyze these surfaces computing their Chern invariants, and proving that they are not fibred over curves of genus greater than one.

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