Showing papers in "Rendiconti Del Circolo Matematico Di Palermo in 2017"
TL;DR: In this paper, the authors provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples, focusing here on ideals with zero dimensional support.
Abstract: The purpose of this note is to provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples. We focus here on ideals with zero dimensional support. This is an area of ongoing active research. We conclude the note with a list of potential promising paths of further research.
48 citations
TL;DR: In this paper, the authors studied the variation of the Newton-Okounkov bodies with respect to the field of rational functions of a smooth projective algebraic surface and a big divisor.
Abstract: 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 O, in the sense that there is a sequence of blowups $$X' \rightarrow X$$
, mapping the smooth, irreducible rational curve $$E\subset X'$$
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 $$X=\mathbb {P}^2$$
.
24 citations
TL;DR: In this article, the authors address the following kind of question: let X be a smooth, irreducible, projective surface and D be a divisor on X satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable?
Abstract: In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and \(D\) a divisor on X satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable? We describe some examples, discuss some conjectures and prove some results that suggest that the answer should in general be negative, unless one puts some really strong hypotheses either on \(D\) or on \(X\).
9 citations
TL;DR: In this paper, locally free sheaves are classified in terms of stability, stability, and rank 2 on a smooth quartic surface, such that the stability of the sheaves is evaluated.
Abstract: Let $$F\subseteq {\mathbb {P}^{3}}$$
be a smooth quartic surface and let $${\mathcal {O}}_F(h):={\mathcal {O}}_{{\mathbb {P}^{3}}}(1)\otimes {\mathcal {O}}_F$$
. In the present paper we classify locally free sheaves $${\mathcal {E}}$$
of rank 2 on F such that $$c_1({\mathcal {E}})={\mathcal {O}}_F(2h), c_2({\mathcal {E}})=8$$
and $$h^1\big (F,{\mathcal {E}}(th)\big )=0$$
for $$t\in \mathbb {Z}$$
. We also deal with their stability.
8 citations
TL;DR: In this article, the authors classify prime order isogenies between algebraic K3 surfaces whose rational transcendental Hodge structures are not isometric, and show that the morphisms induced by these isogenic structures are correspondences by algebraic classes on the product fourfolds; however, they do not satisfy the hypothesis of the Mukai-Nikulin theorem.
Abstract: We classify prime order isogenies between algebraic K3 surfaces whose rational transcendental Hodge structures are not isometric. The morphisms of Hodge structures induced by these isogenies are correspondences by algebraic classes on the product fourfolds; however, they do not satisfy the hypothesis of the well-known Mukai–Nikulin theorem. As an application we describe isogenies obtained from K3 surfaces with an action of a symplectic automorphism of prime order.
8 citations
TL;DR: In this paper, it was shown that the canonical system is base point free and yields an embedding for a family of surfaces of general type S of degree π = 24, p_g (S) = 6, q(S)= 0.
Abstract: We consider a family of surfaces of general type S with \(K_S\) ample, having \(K^2_S = 24, p_g (S) = 6, q(S)=0\). We prove that for these surfaces the canonical system is base point free and yields an embedding \(\Phi _1 : S \rightarrow \mathbb {P}^5\). This result answers a question posed by Kapustka and Kapustka (Bilinkage in codimension 3 and canonical surfaces of degree 18 in \({\mathbb {P}}^5\). arXiv:1312.2824, 2015). We discuss some related open problems, concerning also the case \(p_g(S) = 5\), where one requires the canonical map to be birational onto its image.
6 citations
TL;DR: In this article, the authors studied maximal families W of the Hilbert scheme of smooth connected space curves whose general curve C lies on a smooth degree-s surface S containing a line.
Abstract: We continue the study of maximal families W of the Hilbert scheme, \( {{\mathrm{H}}}(d,g)_{sc}\), of smooth connected space curves whose general curve C lies on a smooth degree-s surface S containing a line. For \(s \ge 4\), we extend the two ranges where W is a unique irreducible (resp. generically smooth) component of \( {{\mathrm{H}}}(d,g)_{sc}\). In another range, close to the boarder of the nef cone, we describe for \(s=4\) and 5 components W that are non-reduced, leaving open the non-reducedness of only 3 (resp. 2) families for \(s \ge 6\) (resp. \(s=5\)), thus making progress to recent results of Kleppe and Ottem in [28]. For \(s=3\) we slightly extend previous results on a conjecture of non-reduced components, and in addition we show its existence in a subrange of the conjectured range.
5 citations
TL;DR: In this paper, the facets of the effective cones of divisors on the blow-up of a divisor in up to five lines in general position were computed and it was shown that up to six lines these threefolds are weak Fano and hence Mori dream spaces.
Abstract: We compute the facets of the effective cones of divisors on the blow-up of $${\mathbb {P}}^3$$
in up to five lines in general position. We prove that up to six lines these threefolds are weak Fano and hence Mori Dream Spaces.
4 citations
TL;DR: In this article, the authors consider the union of certain irreducible components of cohomological support loci of the canonical bundle, which they call standard, and prove a structure theorem about them and single out some particular cases.
Abstract: We consider the union of certain irreducible components of cohomological support loci of the canonical bundle, which we call standard. We prove a structure theorem about them and single out some particular cases, recovering and improving results of Beauville and Chen–Jiang. Finally, as an example of application, we extend to compact Kahler manifolds the classification of smooth complex projective varieties with \(p_1(X)=1\), \(p_3(X)=2\) and \(q(X)=\dim X\).
3 citations
2 citations
TL;DR: In this paper, the authors give a description of the first order deformation space of a regular embedding of a reduced algebraic scheme, and compare their result with results of Ran (in particular, Deformation of Maps, Algebraic Curves and Projective Geometry).
Abstract: In this paper we give a description of the first order deformation space of a regular embedding \(X\hookrightarrow Y\) of reduced algebraic schemes. We compare our result with results of Ran (in particular, Deformation of Maps, Algebraic Curves and Projective Geometry (Trento, 1988), 246–253, Lecture Notes in Math, vol. 1389. Springer, Berlin, 1989, Prop. 1.3).
TL;DR: In this paper, the authors investigate how far the converse is from being true in the case of surfaces and show that two numerically equivalent ample line bundles on a smooth complex projective variety lead to the same Hilbert curve.
Abstract: Given a smooth complex projective variety X and an ample line bundle \(\mathcal{L}\) on it, one can associate the Hilbert curve with \((X,\mathcal{L})\). This is a plane affine algebraic curve, whose equation depends only on the numerical characters of X and \(\mathcal{L}\). In particular, two numerically equivalent ample line bundles on X lead to the same Hilbert curve. Focusing on the case of surfaces, in this paper we investigate how far the converse is from being true.
TL;DR: In this paper, it was shown that if and only if d is even, the linear system embeds S in a smooth rational normal scroll of dimension 3, where S is linearly equivalent to Q, where Q is a quadric on T.
Abstract: Let \((S,{\mathcal {L}})\) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle \({\mathcal {L}}\) of degree \(d > 35\). In this paper we prove that \(K^2_S\ge -d(d-6)\). The bound is sharp, and \(K^2_S=-d(d-6)\) if and only if d is even, the linear system \(|H^0(S,{\mathcal {L}})|\) embeds S in a smooth rational normal scroll \(T\subset {\mathbb {P}}^5\) of dimension 3, and here, as a divisor, S is linearly equivalent to \(\frac{d}{2}Q\), where Q is a quadric on T.
TL;DR: In this paper, the authors studied the characterizations of ample and big divisors for non-integral divisor via the corresponding line bundles and showed that these characterizations can be extended to non-Integral divi corsors via line bundles.
Abstract: Given a Weil non-integral divisor D, it is natural to associate it the line bundle of its integral part $$\mathcal {O}_X([D])$$
. In this work we study which of the classical characterizations of ample and big divisors can be extended to non-integral divisors via the corresponding line bundles.
TL;DR: In this article, the natural action of PGL(V) on the variety of smooth arcs in the projective space has been studied, and a 3-slice algorithm has been proposed to construct a ρ-approximation.
Abstract: We construct a \(\mathbb {Z}\)/3\(\mathbb {Z}\)-slice to the natural action of \({ PGL}(V)\) on the variety of smooth arcs in the projective space \(\mathbb {P}(V)\).
TL;DR: In this paper, a short note written on the occasion of Ph.D. Ellia's 60th birthday is two-fold: (1) to explicitly construct linear determinantal curves meeting transversally along a 0-dimensional compressed Gorenstein set of points with socle degree, and (2) to associate to these curves a rank 2 instanton bundle with quantum number t.
Abstract: This short note written on the occasion of Ph. Ellia’s 60th birthday is two-fold: (1) to explicitly construct linear determinantal curves \(C_t, D_t\subset \mathbb {P}^3\) meeting transversally along a 0-dimensional compressed Gorenstein set of points \(G_{b} \) with socle degree \(b\le 2t-2\), and (2) to associate to \(C_t\), \(D_t\) and \(G_{2t-3}\) a rank 2 instanton bundle \({\mathcal {E}}_t\) on \(\mathbb {P}^3\) with quantum number t.
TL;DR: Boucksom and Nystrom as discussed by the authors presented a presentation of two functions carrying geometric information recently defined on the Newton-Okounkov body, and much can also be found in the Seminaire Bourbaki talk.
Abstract: This note is a presentation of two functions carrying geometric information recently defined on the Newton–Okounkov body by Boucksom and Chen (Compos Math 147:1205–1229, 2011) and Nystrom (Ann Sci EC Norm Super (4) 47:1111–1161, 2014). None of the material presented here is original, and much can also be found in Boucksom’s (Seminaire Bourbaki, 2012) Bourbaki talk.