Search

We consider the Fano scheme $F_k(X)$ of $k$--dimensional linear subspaces contained in a complete intersection $X \subset \mathbb{P}^n$ of multi--degree $\underline{d} = (d_1, \ldots, d_s)$. 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 $\Pi_{i=1}^s d_i > 2$ and we find conditions on $n$, $\underline{d}$ 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 $\Pi_{i=1}^s d_i = 2$., Comment: Referee's suggestions added. To appear on Archiv der Matematik

Let $|L|$ be a linear system on a smooth complex Enriques surface $S$ whose general member is a smooth and irreducible curve of genus $p$, with $L^ 2>0$, and let $V_{|L|, \delta} (S)$ be the Severi variety of irreducible $\delta$-nodal curves in $|L|$. We denote by $\pi:X\to S$ the universal covering of $S$. In this note we compute the dimensions of the irreducible components $V$ of $V_{|L|, \delta} (S)$. In particular we prove that, if $C$ is the curve corresponding to a general element $[C]$ of $V$, then the codimension of $V$ in $|L|$ is $\delta$ if $\pi^{-1}(C)$ is irreducible in $X$ and it is $\delta-1$ if $\pi^ {-1}(C)$ consists of two irreducible components., Comment: Final version incorporating referee's corrections. To appear in the volume dedicated to the BGMS Conference 2018 of the Springer Indam Series

Let $D$ be a very general curve of degree $d=2\ell-\epsilon$ in $\mathbb{P}^2$, with $\epsilon\in \{0,1\}$. Let $\Gamma \subset \mathbb{P}^2$ be an integral curve of geometric genus $g$ and degree $m$, $\Gamma \neq D$, and let $\nu: C\to \Gamma$ be the normalization. Let $\delta$ be the degree of the \emph{reduction modulo 2} of the divisor $\nu^*(D)$ of $C$. In this paper we prove the inequality $4g+\delta\geqslant m(d-8+2\epsilon)+5$. We compare this with similar inequalities due to Geng Xu and Xi Chen. Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces., Comment: to appear in Cont. Math. ("Selim Krein Centennial"), pp. 1-19. Collaboration has benefitted by "MIUR Excellence Department Project" Dep. of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006, grant 346300 for "IMPAN" from the "Simons Foundation"(code: BCSim-2018-s09), funds "Mission Sustainability 2017 - Fam Curves" CUP E81-18000100005 (Tor Vergata)

In this paper we give a description of the first order deformation space of a regular embedding of reduced algebraic schemes. We compare our result with results of Ran (in particular [Ran, Prop. 1.3])., Comment: Added errata corrige. To appear in Rend. Circ. Mat. Palermo, II. Ser

We consider, under suitable assumptions, the following situation: $\mathcal B$ is a component of the moduli space of polarized surfaces and $\mathcal V_{m,\delta}$ is the universal Severi variety over $\mathcal B$ parametrizing pairs $(S,C)$, with $(S,H)\in \mathcal B$ and $C\in |mH|$ irreducible with exactly $\delta$ nodes as singularities. The moduli map $\mathcal V\to \mathcal M_g$ of an irreducible component $\mathcal V$ of $\mathcal V_{m,\delta}$ is generically of maximal rank if and only if certain cohomology vanishings hold. Assuming there are suitable semistable degenerations of the surfaces in $\mathcal B$, we provide sufficient conditions for the existence of an irreducible component $\mathcal V$ where these vanishings are verified. As a test, we apply this to $K3$ surfaces and give a new proof of a result recently independently proved by Kemeny and by the present authors., Comment: submitted paper

Given a smooth projective algebraic surface X, a point O in X and a big divisor D on X, we consider the set of all Newton-Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E,p) which is infinitely near to O, in the sense that there is a sequence of blowups mapping the smooth, irreducible rational curve E to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton-Okounkov bodies as (E, p) varies, focusing on the case X = P2., Comment: 33 pages, 3 figures

In this paper we discuss some variations of Nagata's conjecture on linear systems of plane curves. The most relevant concerns non-effectivity (hence nefness) of certain rays, which we call \emph{good rays}, in the Mori cone of the blow-up $X_n$ of the plane at $n\ge 10$ general points. Nagata's original result was the existence of a good ray for $X_n$ with $n\ge 16$ a square number. Using degenerations, we give examples of good rays for $X_n$ for all $n\ge 10$. As with Nagata's original result, this implies the existence of counterexamples to Hilbert's XIV problem. Finally we show that Nagata's conjecture for $n\le 89$ combined with a stronger conjecture for $n=10$ implies Nagata's conjecture for $n\ge 90$., Comment: 13 pages

In this paper we study the Hilbert scheme of smooth, linearly normal, special scrolls under suitable assumptions on degree, genus and speciality., Comment: Latex2e, shorter version

In this paper we study the Brill-Noether theory of sub-line bundles of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with hyperplane sections isomorphic to C. Most of our results are based on degeneration techniques., Comment: 16 pages, Latex2e, Submitted preprint

This paper is the second in a series. The first one describes pillow degenerations of a $K3$ surface with genus $g$. In this paper we study the $(2,2)$-pillow degeneration of a non-prime $K3$ surface and the braid monodromy of the branch curve of the surface with respect to a generic projection onto $\C\P^2$. In future papers we study the fundamental group of the complement of the branch curve and the fundamental group of the Galois cover of the surface with respect to this generic projection., Comment: 31 pages

We classify all irreducible projective threefolds $X$ which are $k$-defective, i.e. some $k$-secant variety of $X$ has dimension less than the expected value. This results extends the classical Scorza's classification of the case $k=1$., Comment: AMSLaTeX, 30 pages

In this paper we study some properties of degenerations of surfaces whose general fibre is a smooth projective surface and whose central fibre is a reduced, connected surface $X \subset IP^r$, $r \geq 3$, which is assumed to be a union of smooth projective surfaces, in particular of planes. Our original motivation has been a series of papers of G. Zappa which appeared in the 1940-50's regarding degenerations of scrolls to unions of planes. Here, we present a first set of results on the subject; other aspects are still work in progress and will appear later. We first study the geometry and the combinatorics of a surface like $X$, considered as a reduced, connected surface on its own; then we focus on the case in which X is the central fibre of a degeneration of relative dimension two over the complex unit disk. In this case, we deduce some of the intrinsic and extrinsic invariants of the general fibre from the ones of its central fibre. In the particular case of $X$ a central fibre of a semistable degeneration, i.e. $X$ has only global normal crossing singularities and the total space of the degeneration is smooth, some of the above invariants can be also computed by topological methods (i.e., the Clemens-Schmid exact sequence). Our results are more general, not only because the computations are independent on the fact that $X$ is the central fibre of a degeneration, but also because the degeneration is not semistable in general., Comment: latex2e, 26 pages, 11 figures

The number of apparent double points of a smooth, irreducible projective variety $X$ of dimension $n$ in $\Proj^{2n+1}$ is the number of secant lines to $X$ passing through the general point of $\Proj^{2n+1}$. This classical notion dates back to Severi. In the present paper we classify smooth varieties of dimension at most three having one apparent double point. The techniques developed for this purpose allow to treat a wider class of projective varieties., Comment: 31 pages, AMSLaTeX2e, to appear in J.A.G

Let X be a smooth complex surface of general type such that the image of the canonical map $\phi$ of X is a surface $\Sigma$ and that $\phi$ has degree $\delta\geq 2$. Let $\epsilon\colon S\to \Sigma$ be a desingularization of $\Sigma$ and assume that the geometric genus of S is not zero. Beauville has proved that in this case S is of general type and $\epsilon$ is the canonical map of S. Beauville has also constructed the only infinite series of examples $\phi:X\to \Sigma$ with the above properties that was known up to now. Starting from his construction, we define a {\em good generating pair}, namely a pair $(h:V\to W, L)$ where h is a finite morphism of surfaces and L is a nef and big line bundle of W satisfying certain assumptions. We show that by applying a construction analogous to Beauville's to a good generating pair one obtains an infinite series of surfaces of general type whose canonical map is 2-to-1 onto a canonically embedded surface. In this way we are able to construct more infinite series of such surfaces. In addition, we show that good generating pairs have bounded invariants and that there exist essentially only 2 examples with $\dim |L|>1$. The key fact that we exploit for obtaining these results is that the Albanese variety P of V is a Prym variety and that the fibre of the Prym map over P has positive dimension., Comment: 40 pages, LaTeX 2.09

One of the simplest examples of a smooth, non degenerate surface in P^4 is the quintic elliptic scroll. It can be constructed from an elliptic normal curve E by joining every point on E with the translation of this point by a non-zero 2-torsion point. The same construction can be applied when E is replaced by a (lineaerly normally embedded) abelian variety A. In this paper we ask the question when the resulting scroll Y is smooth. If A is an abelian surface embedded by a line bundle L of type (d_1,d_2) and r=d_1d_2, then we prove that for general A the scroll Y is smooth if r is at least 7 with the one exception where r=8 and the 2-torsion point is in the kernel K(L) of L. In this case Y is singular.The case r=7 is particularly interesting, since then Y is a smooth threefold in P^6 with irregularity 2. The existence of this variety seems not to have been noticed before. One can also show that the case of the quintic elliptic scroll and the above case are the only possibilities where Y is smooth and the codimension of Y is at most half the dimension of the surrounding projective space., Comment: 23 pages, Plain Tex. Some corrections made. To appear: Annali Sc. Norm. Sup. di Pisa

In this article we address the problem of computing the dimension of the space of plane curves of degree $d$ with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple $(-1)$-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed in a previous article ("Degenerations of Planar Linear Systems", alg-geom/9702015) to show that the conjecture holds for all $m \leq 12$., Comment: 15 pages

The Severi variety V_{n,d} of a smooth projective surface S is defined as the subvariety of the linear system |O_S(n)|, which parametrizes curves with d nodes. We show that, for a general surface S of degree k in P^3 and for all n>k-1, d=0,...,dim(|O_S(n)|), there exists one component of V_{n,d} which is reduced, of the expected dimension dim(|O_S(n)|)-d. Components of the expected dimension are the easiest to handle, trying to settle an enumerative geometry for singular curves on surfaces. On the other hand, we also construct examples of reducible Severi varieties, on general surfaces of degree k>7 in P^3., Comment: AMSTeX, AMSppt style, 14 pages

The union of two quintic elliptic scrolls in P^4 intersecting transversally along an elliptic normal quintic curve is a singular surface Z which behaves numerically like a bielliptic surface. In the appendix to the paper [W. Decker et al.: Syzygies of abelian and bielliptic surfaces in P^4, alg-geom/9606013] where the equations of this singular surface were computed, we proved that Z defines a smooth point in the appropriate Hilbert scheme and that Z cannot be smoothed in P^4. Here we consider the analogous situation in higher dimensional projective spaces P^{n-1}, where, to our surprise, the answer depends on the dimension n-1. If n is odd the union of two scrolls cannot be smoothed, whereas it can be smoothed if n is even. We construct an explicit smoothing., Comment: LaTeX2e with theorem, amstex, amssymb, amscd packages; 11 pages

Fixing $n$ general points $p_i$ in the plane, what is the dimension of the space of plane curves of degree $d$ having multiplicity $m_i$ at $p_i$ for each $i$? In this article we propose an approach to attack this problem, and demonstrate it by successfully computing this dimension for all $n$ and for $m_i$ constant, at most 3. This application, while previously known (see \cite{hirschowitz1}), demonstrates the utility of our approach, which is based on an analysis of the corresponding linear system on a degeneration of the plane itself, leading to a simple recursion for these dimensions. We also obtain results in the ``quasi-homogeneous'' case when all the multiplicities are equal except one; this is the natural family to consider in the recursion., Comment: material is streamlined and some is moved to a forthcoming paper

Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The purpose of this article is to further develop a method to classify varieties having $C$ as their curve section, using Gaussian map computations. In a previous article a careful analysis of the degeneration to the cone over the hyperplane section was made for _prime_ Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index (which still have Picard number one). We are also able to classify Mukai varieties, i.e. varieties of dimension four or more with canonical curve sections., Comment: 22 pages, fairly plain TeX; in the corrigendum, added below the paper, we correct a mistake in the statement and proof of Lemma 2.3(d)

Let $S_{g}$ be a general prime K3 surface in $P^g$ of genus $g \geq 3$ or a general double cover of $P^2$ ramified along a sextic curve for $g = 2$ and $S = S_{i,g}$ its {\it i}-th Veronese embedding. In this article we compute the corank of the Gaussian map $\Phi:\bigwedge^2 H^0(S,O_S(1)) \to H^0(S,\Omega_S^1(2))$ for $i \geq 2, g \geq 2$ and $i=1, g \geq 17$. The main idea is to reduce the surjectivity of $\Phi$ to an application of the Kawamata-Viehweg vanishing theorem on the blow-up of $S \times S$ along,the diagonal. This is seen to apply once the hyperplane divisor of the K3 surface $S$ can be decomposed as a sum of three suitable birationally ample divisors. We show that such a decomposition exists when $i \geq 3$ or on some K3 surfaces, constructed using the surjectivity of the period mapping, when $i = 1, g \geq 17$ or $i=2, g \geq 7$., Comment: 21 pages, fairly plain TeX