Two general multiple planes having the same branch curve cannot be too different; a central result in the theory of multiple planes first proved by Chisini asserts that two such multiple planes, with some additional hypothesis, are... more
Two general multiple planes having the same branch curve cannot be too different; a central result in the theory of multiple planes first proved by Chisini asserts that two such multiple planes, with some additional hypothesis, are birational. Let S be a complex projective nonsingular algebraic surface, R a net on S, f:S \to P^2 che associated multiple plane. We prove that if the moving divisor of R is ample, then the ramification curve G of f is ample too. So S \setminus G is Stein. Now, let f:S \to P^2, f':S'\to P^2 be two general multiple planes having the same branch curve and such that the moving divisors of the corresponging nets are ample. Then one can extend to S and S' an isomorphism between two tubular neighbourhoods of the ramification curves G and G', whose existence was claimed by Chisini
Let X be a complex projective smooth threefold and let L be an ample line bundle on X, spanned by its global sections. Pairs (X,L) as above with c_1(L)^3=3, g(L)=3 and h^0(L)=4 are classified by combining the investigation of their... more
Let X be a complex projective smooth threefold and let L be an ample line bundle on X, spanned by its global sections. Pairs (X,L) as above with c_1(L)^3=3, g(L)=3 and h^0(L)=4 are classified by combining the investigation of their adjunction theoretic sctructure with the classification of surfaces polarized by an ample and spanned line bundle of genus three
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I report on work done in the last few years on ample vector bundles and special varieties and on some aspects of a work in progress on ample vector bundles and surfaces with a trigonal curve section, offering a connection with the study... more
I report on work done in the last few years on ample vector bundles and special varieties and on some aspects of a work in progress on ample vector bundles and surfaces with a trigonal curve section, offering a connection with the study of homaloidal nets of plane curve made by Cremona. Finally some ideas concerning special varieties as ample subvarieties are discusse
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ABSTRACT Let Y be a Fano manifold of dimension n ⩾ 3 with b2(Y and index n – 1 and let A be a projective manifold which is a double cover of Y. We determine which complex projective manifolds can admit A among their hyperplane sections.
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Research Interests: Mathematics, Combinatorics, Pure Mathematics, Genus, Dimensional, and 3 moreFolds, Hyperplane, and Scroll
Preface - Lucian Baxdescu and Michael Schneider: Formal functions, connectivity and homogeneous spaces - Luca Barbieri-Viale: On algebraic 1-motives related to Hodge cycles - Arnaud Beauville: The Szpiro inequality for higher genus... more
Preface - Lucian Baxdescu and Michael Schneider: Formal functions, connectivity and homogeneous spaces - Luca Barbieri-Viale: On algebraic 1-motives related to Hodge cycles - Arnaud Beauville: The Szpiro inequality for higher genus fibrations - Giuseppe Borrelli: On regular surfaces of general type with pg=2 and non-birational bicanonical map - Fabricio Catanese and Frank-Olaf Schreyer: Canonical projections of irregular algebraic surfaces - Ciro Ciliberto and Margarida Mendes Lopes: On surfaces with pg=2, q=1 and non-birational bicanonical map - Alberto Conte, Marina Marchisio and Jacob Murre: On unirationality of double covers of fixed degree and large dimension a method of Ciliberto - Alessio Corti and Miles Reid: Weighted Grassmannians - Tommaso de Fernex and Lawrence Ein: Resolution of indeterminacy of pairs - Vladimir Guletskiix and Claudio Pedrini: The Chow motive of the Godeaux surface - Yujiro Kawamata: Francia's flip and derived categories - Kazuhiro Konno: On the quadric huil of a canonical surface - Adrian Langer: A note on Bogomolov's instability and Higgs sheaves - Antonio Lanteri and Raquel Mallavibarrena: Jets of antimulticanonical bundles on Del Pezzo surfaces of degree no.2 - Margarida Mendes Lopes and Rita Pardini: A survey on the bicanonical map of surfaces with pg=0 and K2 2 - Francesco Russo: The antibirational involutions of the plane and the classification of real del Pezzo surfaces - Vyacheslav V. Shokurov: Letters of a birationalist IV. Geometry of log flips - Andrew J. Sommese, Jean Verschelde and Charles Wampler A method for tracking singular paths with application to the numerical irreducible decomposition.
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Research Interests: Mathematics, Pure Mathematics, Dead Sea Scrolls, Cohomology, Jets, and 3 moreScrolls, Hyperplane, and Scroll
Let X be a smooth complex projective variety endowed with an ample vector bundle E admitting a global section whose zero locus is a smooth subvariety Z of the expected dimension and let H be an ample line bundle on X, whose restriction... more
Let X be a smooth complex projective variety endowed with an ample vector bundle E admitting a global section whose zero locus is a smooth subvariety Z of the expected dimension and let H be an ample line bundle on X, whose restriction H_Z to Z is very ample. Triplets (X,E,H) are studied and classified under the assumption that the delta genus of (Z,H_Z) is either small (\leq 3) or small in comparison with the corank of E or the degree
The author remembers Carlo Felice Manara in the period ranging from the late Sixties to the early Eighties through memories of his apprenticeship. The paper describes some ideas about multiple planes and their branch curves, this being... more
The author remembers Carlo Felice Manara in the period ranging from the late Sixties to the early Eighties through memories of his apprenticeship. The paper describes some ideas about multiple planes and their branch curves, this being the main theme addressed by Chisini and his School in Milan, where Manara was a prominent figure. The discussion includes an outline of recent developements and an attempt to put the subject in the general framework of the geometry of linear systems
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We prove that any nonsingular geometrically ruled surface X of irregularity q=1 embedded in P^4 is an elliptic scroll of degree 5. We show that the algebraic system of the fundamental sections of this surface is isomorphic to the elliptic... more
We prove that any nonsingular geometrically ruled surface X of irregularity q=1 embedded in P^4 is an elliptic scroll of degree 5. We show that the algebraic system of the fundamental sections of this surface is isomorphic to the elliptic base curve, and we describe a geometric construction of the scroll
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Under some positivity assumptions, extension properties of rationally con- nected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering... more
Under some positivity assumptions, extension properties of rationally con- nected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applica- tions of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration.
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Let L be an ample line bündle on a complex connected projective manifold X. Assume that L is spanned by a vector space F<= H°(L) of global sections. Let \V\ stand for the linear system corresponding to the elements of V and let 2 (V)... more
Let L be an ample line bündle on a complex connected projective manifold X. Assume that L is spanned by a vector space F<= H°(L) of global sections. Let \V\ stand for the linear system corresponding to the elements of V and let 2 (V) denote the discriminant locus of F, i.e., the set of points of | V\ corresponding to singular divisors. If | V\ gives an embedding it is well known that ® (V) is irreducible and that for any xe X there are smooth elements of | V \ containing x. For V which merely Spans L (even if L is very ample), these results in general fail.
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Let ε be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section s ∈ Γ(ε) whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X-r:= n - r. Assume that Z is not... more
Let ε be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section s ∈ Γ(ε) whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X-r:= n - r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.
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We investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.
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Let S be a smooth surface embedded in a projective space, whose general osculating space has the expected dimension. Inside the dual variety of S one can consider the second discriminant locus, which parametrizes the hyperplane sections... more
Let S be a smooth surface embedded in a projective space, whose general osculating space has the expected dimension. Inside the dual variety of S one can consider the second discriminant locus, which parametrizes the hyperplane sections of S having some singular point of multiplicity ≥3. In this paper the various components of the second discriminant loci of Del Pezzo surfaces are investigated from a unifying point of view. This allows us to describe the second dual varieties of such surfaces and to understand their singular loci.
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The bad locus of a free linear system ℒ on a normal complex projective variety X is defined as the set B( ℒ ) ⊆ X of points that are not contained in any irreducible and reduced member of ℒ. In this paper we provide a geometric... more
The bad locus of a free linear system ℒ on a normal complex projective variety X is defined as the set B( ℒ ) ⊆ X of points that are not contained in any irreducible and reduced member of ℒ. In this paper we provide a geometric description of such locus in terms of the morphism defined by ℒ. In particular, assume that dim X ≥ 2 and ℒ is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(ℒ) is empty unless X is a surface. Then we prove that, when the latter occurs, B(ℒ) is not empty if and only if ℒ defines a morphism onto a two dimensional cone, in which case B(ℒ) is the inverse image of the vertex of the cone.
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A contribution to the classification of complex projective manifolds admitting a smooth triple cover of the projective space among their hyperplane sections is given
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The classical subject of surfaces containing a hyperelliptic curve (here a double cover of P1 ) among their hyperplane sections was settled some years ago by the third author and Van de Ven [SV] (see also [Se], [Io]). This paper is... more
The classical subject of surfaces containing a hyperelliptic curve (here a double cover of P1 ) among their hyperplane sections was settled some years ago by the third author and Van de Ven [SV] (see also [Se], [Io]). This paper is devoted to answering the following general question arising very naturally from that problem.
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ABSTRACT Let (X,L) be a smooth polarized variety of dimension n. Let A∈|L| be an irreducible hypersurface and let Σ be the singular locus of A. We assume that Σ is a smooth subvariety of dimension k≥2, and odd codimension ≥3. Motivated... more
ABSTRACT Let (X,L) be a smooth polarized variety of dimension n. Let A∈|L| be an irreducible hypersurface and let Σ be the singular locus of A. We assume that Σ is a smooth subvariety of dimension k≥2, and odd codimension ≥3. Motivated from the results of Beltrametti et al. (J. Math. Soc. Jpn. 2014), we study the nefness and bigness of the adjoint bundle K Σ +(k-2)L Σ in this framework. Several explicit examples show that the results are effective.
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Let X subset of P-N be a scroll over an m-dimensional variety Y. We find the locally free sheaves on X governing the osculating behavior of X, and, under certain dimension assumptions, we compute the cohomology class and the degree of the... more
Let X subset of P-N be a scroll over an m-dimensional variety Y. We find the locally free sheaves on X governing the osculating behavior of X, and, under certain dimension assumptions, we compute the cohomology class and the degree of the inflectional locus of X. The case m = 1 was treated in [9]. Here we treat the case m >= 2, which is more complicated for at least two reasons: the expression for the osculating sheaves and the computations of the class of the inflectional locus become more complex, and the dimension requirements needed to ensure validity of the formulas are more severe.
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We investigate higher order dual varieties of projective manifolds whose osculatory behavior is the best possible. In particular, for a k-jet ample surface we prove the nondegeneratedness of the k-th dual variety and for 2-regular... more
We investigate higher order dual varieties of projective manifolds whose osculatory behavior is the best possible. In particular, for a k-jet ample surface we prove the nondegeneratedness of the k-th dual variety and for 2-regular surfaces we investigate the degree of the second dual variety.
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ABSTRACT Some progress in the study of smooth complex projective ruled surfaces SPNwith low class m is obtained by means of adjunction theory. In particular surfaces of degree d and class m ≤ 2d + 2 are completely classified and the... more
ABSTRACT Some progress in the study of smooth complex projective ruled surfaces SPNwith low class m is obtained by means of adjunction theory. In particular surfaces of degree d and class m ≤ 2d + 2 are completely classified and the result is extended to higher dimensions.
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We prove that, if a smooth complex projective surface S P N is k-regular, then its k-th order dual variety has the expected dimension, except if S is the k-th Veronese surface. This answers positively a conjecture stated in a previous... more
We prove that, if a smooth complex projective surface S P N is k-regular, then its k-th order dual variety has the expected dimension, except if S is the k-th Veronese surface. This answers positively a conjecture stated in a previous paper.