Papers by Francesco Bastianelli
Transactions of the American Mathematical Society, Aug 5, 2019
In this paper we investigate the cone Pseffn(C d) of pseudoeffective n-cycles in the symmetric pr... more In this paper we investigate the cone Pseffn(C d) of pseudoeffective n-cycles in the symmetric product C d of a smooth curve C. We study the convex-geometric properties of the cone Dn(C d) generated by the n-dimensional diagonal cycles. In particular we determine its extremal rays and we prove that Dn(C d) is a perfect face of Pseffn(C d) along which Pseffn(C d) is locally finitely generated.
arXiv (Cornell University), Nov 21, 2017
In this paper we investigate the cone Pseffn(C d) of pseudoeffective n-cycles in the symmetric pr... more In this paper we investigate the cone Pseffn(C d) of pseudoeffective n-cycles in the symmetric product C d of a smooth curve C. We study the convex-geometric properties of the cone Dn(C d) generated by the n-dimensional diagonal cycles. In particular we determine its extremal rays and we prove that Dn(C d) is a perfect face of Pseffn(C d) along which Pseffn(C d) is locally finitely generated.
Bollettino dell'Unione Matematica Italiana, Jan 8, 2018
Geometriae Dedicata, Apr 19, 2019
Consider the Fano scheme F k (Y) parameterizing k-dimensional linear subspaces contained in a com... more Consider the Fano scheme F k (Y) parameterizing k-dimensional linear subspaces contained in a complete intersection Y ⊂ P m of multi-degree d = (d 1 ,. .. , ds). It is known that, if t := s i=1 d i +k k − (k + 1)(m − k) 0 and Π s i=1 d 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 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 ] ∈ W d,k the scheme F k (Y) is zero-dimensional of length one. This implies that W d,k is rational.
arXiv (Cornell University), Dec 17, 2009
Let X be a complex projective variety and consider the morphism ψ k : k H 0 (X, Ω 1 X) −→ H 0 (X,... more Let X be a complex projective variety and consider the morphism ψ k : k H 0 (X, Ω 1 X) −→ H 0 (X, Ω k X). We use Galois closures of finite rational maps to introduce a new method for producing varieties such that ψ k 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 g ≥ 2. The topological index of these surfaces is negative and this provides a counterexample to a conjecture on Lagrangian surfaces formulated in [3].
Mathematische Zeitschrift, Oct 30, 2014
A point p ∈ C on a smooth complex projective curve of genus g ≥ 3 is subcanonical if the divisor ... more A point p ∈ C on a smooth complex projective curve of genus g ≥ 3 is subcanonical if the divisor (2g − 2)p is canonical. The subcanonical locus Gg ⊂ Mg,1 described by pairs (C, p) as above has dimension 2g−1 and consists of three irreducible components. Apart from the hyperelliptic component G hyp g , the other components G odd g and G even g 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 G r g of pairs (C, p) in Gg such that h 0 (C, (g − 1)p) ≥ r + 1 and h 0 (C, (g − 1)p) ≡ r + 1 (mod 2). In particular, we provide a lower bound on their dimension, and we prove its sharpness for r ≤ 3. 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.
Mathematische Nachrichten
Given a smooth hypersurface of degree , we study the cones swept out by lines having contact orde... more Given a smooth hypersurface of degree , we study the cones swept out by lines having contact order at a point . In particular, we prove that if X is general, then for any and , the cone has dimension exactly . Moreover, when X is a very general hypersurface of degree , we describe the relation between the cones 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 .
arXiv (Cornell University), Jul 2, 2020
Using the moduli space of semiorthogonal decompositions in a smooth projective family introduced ... more Using the moduli space of semiorthogonal decompositions in a smooth projective family introduced by the second, the third and the fourth author, we discuss indecomposability results for derived categories in families. In particular, we prove that given a smooth projective family of varieties, if the derived category of the general fibre does not admit a semiorthogonal decomposition, the same happens for every fibre of the family. As a consequence, we deduce that in a smooth family of complex projective varieties, if there exists a fibre such that the base locus of its canonical linear series is either empty or finite, then any fibre of the family has indecomposable derived category. Then we apply our results to achieve indecomposability of the derived categories of various explicit classes of varieties, as e.g. n-fold symmetric products of curves (with 0 < n < ⌊(g + 3)/2⌋), Horikawa surfaces, an interesting class of double covers of the projective plane introduced by Ciliberto, and Hilbert schemes of points on these two classes of surfaces.
arXiv (Cornell University), Nov 4, 2015
This short paper is an appendix to [6]. We prove an existence result for families of curves havin... more This short paper is an appendix to [6]. We prove an existence result for families of curves having low gonality, and lying on fundamental loci of first order congruences of lines in 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 ⊂ P n+1 of degree d ≥ 3n − 2 ≥ 7, and a dominant rational map f : X P n , then deg(f) ≥ d − 1, and equality holds if and only if f is the projection from a point of X.
Let X be a complex projective variety and consider the morphism ψ k : k H 0 (X, Ω 1 X) −→ H 0 (X,... more Let X be a complex projective variety and consider the morphism ψ k : k H 0 (X, Ω 1 X) −→ H 0 (X, Ω k X). We use Galois closures of finite rational maps to introduce a new method for producing varieties such that ψ k 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 g ≥ 2. The topological index of these surfaces is negative and this provides a counterexample to a conjecture on Lagrangian surfaces formulated in [3].
This paper is a short summary of the main results in the thesis [1] and in the paper [40]. We dea... more This paper is a short summary of the main results in the thesis [1] and in the paper [40]. We deal throughout with several problems on the surfaces obtained as second symmetric products of smooth projective curves. In particular, we treat both some attempts at extending the notion of gonality for curves and some classical problems, as the study of the ample cone in the Neron-Severi group. Moreover, we develop a family of examples of Lagrangian surfaces having particular topological properties.
Archiv der Mathematik, 2020
We consider the Fano scheme F k (X) of k-dimensional linear subspaces contained in a complete int... more We consider the Fano scheme F k (X) of k-dimensional linear subspaces contained in a complete intersection X ⊂ P n of multi-degree d = (d 1 ,. .. , ds). Our main result is an extension of a result of Riedl and Yang concerning Fano schemes of lines on very general hypersurfaces: we consider the case when X is a very general complete intersection and Π s i=1 d i > 2 and we find conditions on n, d and k under which F k (X) does not contain either rational or elliptic curves. At the end of the paper, we study the case Π s i=1 d i = 2.
Geometriae Dedicata, 2019
Consider the Fano scheme F k (Y) parameterizing k-dimensional linear subspaces contained in a com... more Consider the Fano scheme F k (Y) parameterizing k-dimensional linear subspaces contained in a complete intersection Y ⊂ P m of multi-degree d = (d 1 ,. .. , ds). It is known that, if t := s i=1 d i +k k − (k + 1)(m − k) 0 and Π s i=1 d 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 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 ] ∈ W d,k the scheme F k (Y) is zero-dimensional of length one. This implies that W d,k is rational.
Transactions of the American Mathematical Society, 2019
In this paper we investigate the cone Pseff n ( C d ) \operatorname {Pseff}_n(C_d) of pseudoeff... more In this paper we investigate the cone Pseff n ( C d ) \operatorname {Pseff}_n(C_d) of pseudoeffective n n -cycles in the symmetric product C d C_d of a smooth curve C C . We study the convex-geometric properties of the cone D n ( C d ) \mathcal {D}_n(C_d) generated by the n n -dimensional diagonal cycles. In particular we determine its extremal rays and we prove that D n ( C d ) \mathcal {D}_n(C_d) is a perfect face of Pseff n ( C d ) \operatorname {Pseff}_n(C_d) along which Pseff n ( C d ) \operatorname {Pseff}_n(C_d) is locally finitely generated.
Journal de Mathématiques Pures et Appliquées, 2019
This paper concerns the existence of curves with low gonality on smooth hypersurfaces of sufficie... more 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 ⊂ P n+1 is a hypersurface of degree d n + 2, and if C ⊂ X is an irreducible curve passing through a general point of X, then its gonality verifies gon(C) d − n, and equality is attained on some special hypersurfaces. We prove that if X ⊂ P n+1 is a very general hypersurface of degree d 2n + 2, the least gonality of an irreducible curve C ⊂ X passing through a general point of X is gon(C) = d − √ 16n+1−1 2 , apart from a series of possible exceptions, where gon(C) may drop by one.
Annali di Matematica Pura ed Applicata (1923 -), 2018
Let C be a smooth projective curve of genus g ≥ 2. Fix an integer r ≥ 0, and let k = (k 1 ,. .. ,... more Let C be a smooth projective curve of genus g ≥ 2. Fix an integer r ≥ 0, and let k = (k 1 ,. .. , k n) be a sequence of positive integers with n i=1 k i = g − 1. In this paper, we study n-pointed curves (C, p 1 ,. .. , p n) such that the line bundle L := O C n i=1 k i p i is a theta-characteristic with h 0 (C, L) ≥ r + 1 and h 0 (C, L) ≡ r + 1 (mod 2). We prove that they describe a sublocus G r g (k) of M g,n having codimension at most g − 1 + r (r −1) 2. Moreover, for any r ≥ 0, k as above, and g greater than an explicit integer g(r) depending on r , we present irreducible components of G r g (k) attaining the maximal codimension in M g,n , so that the bound turns out to be sharp. Keywords Theta-characteristic • Moduli of curves • Spin curve • Subcanonical point Mathematics Subject Classification 14H10 • 14H45 • 14H51 This work was partially supported by the national projects FIRB 2012 "Spazi di moduli e applicazioni" and PRIN 2010-11 "Geometry of algebraic varieties" founded by the Ministero dell'Istruzione, dell'Università e della Ricerca, by the project FAR 2010 "Geometria e Topologia" founded by University of Milano-Bicocca, and by Istituto Nazionale di Alta Matematica (GNSAGA).
Journal of Algebra, 2017
The degree of irrationality irr(X) of a n-dimensional complex projective variety X is the least d... more The degree of irrationality irr(X) of a n-dimensional complex projective variety X is the least degree of a dominant rational map X P n. It is a well-known fact that given a product X × 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 ⊂ P 3 of degree d ≥ 5, and we prove that irr(S × P m) = irr(S) for any integer m ≥ 0, whereas irr(Y) < irr(S) occurs for some Y dominating S if and only if S contains a rational curve.
Bollettino dell'Unione Matematica Italiana, 2017
This short paper concerns the existence of curves with low gonality on smooth hypersurfaces X ⊂ P... more This short paper concerns the existence of curves with low gonality on smooth hypersurfaces X ⊂ P n+1. After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome "Tor Vergata" on January 11th-15th, 2016. In particular, we obtained that if X ⊂ P n+1 is a very general hypersurface of degree d 2n + 2, the least gonality of a curve C ⊂ X passing through a general point of X is gon(C) = d − √ 16n+1−1 2 , apart from some exceptions we list. The research leading to these results has received funding from the European Research Council under the
Compositio Mathematica, 2017
We study various measures of irrationality for hypersurfaces of large degree in projective space ... more 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 $X\subseteq \mathbf{P}^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $d\geqslant 2n+1$, then any dominant rational mapping $f:X{\dashrightarrow}\mathbf{P}^{n}$ 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 concerning varieties of multi-secant lines.
Transactions of the American Mathematical Society, 2012
Let C be a smooth complex projective curve of genus g and let C (2) be its second symmetric produ... more Let C be a smooth complex projective curve of genus g and let C (2) be its second symmetric product. This paper concerns the study of some attempts at extending to C (2) the notion of gonality. In particular, we prove that the degree of irrationality of C (2) is at least g − 1 when C is generic and that the minimum gonality of curves through the generic point of C (2) equals the gonality of C. In order to produce the main results we deal with correspondences on the k-fold symmetric product of C, with some interesting linear subspaces of P n enjoying a condition of Cayley-Bacharach type, and with monodromy of rational maps. As an application, we also give new bounds on the ample cone of C (2) when C is a generic curve of genus 6 ≤ g ≤ 8.
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Papers by Francesco Bastianelli