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14
Curves having one place at infinity and linear systems on rational surfaces
 J. Pure Appl. Algebra
"... Abstract. Denoting by Ld(m0, m1,..., mr) the linear system of plane curves passing through r + 1 generic points p0, p1,..., pr of the projective plane with multiplicity mi (or larger) at each pi, we prove the HarbourneHirschowitz Conjecture for linear systems Ld(m0, m1,..., mr) determined by a wide ..."
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Cited by 4 (4 self)
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Abstract. Denoting by Ld(m0, m1,..., mr) the linear system of plane curves passing through r + 1 generic points p0, p1,..., pr of the projective plane with multiplicity mi (or larger) at each pi, we prove the HarbourneHirschowitz Conjecture for linear systems Ld(m0, m1,..., mr) determined by a wide family of systems of multiplicities m = (mi) r i=0 and arbitrary degree d. Moreover, we provide an algorithm for computing a bound of the regularity of an arbitrary system m and we give its exact value when m is in the above family. To do that, we prove an H 1vanishing theorem for line bundles on surfaces associated with some pencils “at infinity”. 1.
NONEMPTINESS OF SYMMETRIC DEGENERACY LOCI
, 2003
"... Abstract. Let V be a renk N vector bundle on a ddimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that S 2 V ∗ ⊗ L is ample. Suppose that the maximum rank of the quadratic form at any point of X is r> 0. The main result of t ..."
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Cited by 1 (1 self)
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Abstract. Let V be a renk N vector bundle on a ddimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that S 2 V ∗ ⊗ L is ample. Suppose that the maximum rank of the quadratic form at any point of X is r> 0. The main result of this paper is that if d> N − r, the locus of points where the rank of the quadratic form is at most r − 1 is nonempty. We give some applications to subschemes of matrices, and to degeneracy loci associated to embeddings in projective space. The paper concludes with an appendix on Gysin maps. The main result of the appendix, which may be of independent interest, identifies a Gysin map with the natural map from ordinary to relative cohomology. 1.
DUAL VARIETIES AND THE DUALITY OF THE SECOND FUNDAMENTAL FORM
, 1997
"... Dedicated to Professor TzeeChar Kuo on his sixtieth birthday Abstract. First, we consider a compact realanalytic irreducible subvariety M in a sphere and its dual variety M ∨. We explain that two matrices of the second fundamental forms for both varieties M and M ∨ can be regarded as the inverse m ..."
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Dedicated to Professor TzeeChar Kuo on his sixtieth birthday Abstract. First, we consider a compact realanalytic irreducible subvariety M in a sphere and its dual variety M ∨. We explain that two matrices of the second fundamental forms for both varieties M and M ∨ can be regarded as the inverse matrices of each other. Also generalization in hyperbolic space is explained. 1. Spherical case In this article I would like to explain main ideas in my recent results on duality of the second fundamental form. (Urabe[6].) Theory of dual varieties in the complex algebraic geometry is very interesting. (Griffiths and Harris [1], Kleiman [2], Piene [4], Urabe [5], Wallace [7].) Let P be a complex projective space of dimension N, and X ⊂ P be a complex algebraic subvariety. The set of all hyperplanes in P forms another projective space P ∨ of dimension N, which is called the dual projective space of P. The dual projective space (P ∨ ) ∨ of P ∨ is identified with P. The closure in P ∨ of the set of tangent
Enumeration of unisingular algebraic hypersurfaces
, 2007
"... We enumerate complex algebraic hypersurfaces in P n, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equisingular strata in the parameter space of hypersurfaces. We suggest an inductive procedure ..."
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We enumerate complex algebraic hypersurfaces in P n, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equisingular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on an intersection theory combined with liftings and degenerations. The procedure computes the (co)homology class in question, whenever a given singularity type is properly defined and the stratum possesses good geometric properties. We consider in detail the generalized Newtonnondegenerate singularities. We also give examples of enumeration in some other cases.
Varieties of clusters and Enriques diagrams
, 2008
"... Given a surface S and an integer r ≥ 1, there is a variety Xr−1 parametrizing all clusters of r proper and infinitely near points of S (see [17]). We study the geometry of the varieties Xr, showing that for every Enriques diagram D of r vertices the subset Cl(D) ⊂ Xr−1 of the clusters with Enriques ..."
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Given a surface S and an integer r ≥ 1, there is a variety Xr−1 parametrizing all clusters of r proper and infinitely near points of S (see [17]). We study the geometry of the varieties Xr, showing that for every Enriques diagram D of r vertices the subset Cl(D) ⊂ Xr−1 of the clusters with Enriques diagram D is locally closed. We study also the relative positions of the subvarieties Cl(D), showing that they do not form a stratification and giving criteria for adjacencies between them.
HODGE GENERA OF ALGEBRAIC VARIETIES, II.
, 2007
"... Abstract. We study the behavior of Hodgetheoretic genera under morphisms of complex algebraic varieties. We prove that the additive χygenus which arises in the motivic context satisfies the socalled “stratified multiplicative property”, which shows how to compute the invariant of the source of a ..."
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Abstract. We study the behavior of Hodgetheoretic genera under morphisms of complex algebraic varieties. We prove that the additive χygenus which arises in the motivic context satisfies the socalled “stratified multiplicative property”, which shows how to compute the invariant of the source of a proper surjective morphism from its values on various varieties that arise from the singularities of the map. By considering morphisms to a curve, we obtain a Hodgetheoretic analogue of the RiemannHurwitz formula. We also consider the contribution of monodromy to the χygenus of a smooth projective family, and prove an AtiyahMeyer formula for twisted χygenera. This formula measures the deviation from multiplicativity of the χygenus, and expresses the correction terms as highergenera associated to cohomology classes of the quotient of the total period domain by the action of the monodromy group. By making use of Saito’s theory of mixed Hodge modules, we also obtain characteristic class formulae of AtiyahMeyer type. In the last section we use intersection homology for the study of twisted Hodgetheoretic genera in the singular setting: we extend some results of [10] regarding the stratified multiplicative property of the Iχygenus, and conjecture a AtiyahMeyer type formula for twisted Iχygenera. Contents
c○1994 American Mathematical Society
, 1994
"... Abstract. This paper announces results on the behavior of some important algebraic and topological invariants — Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. — and their associated characteristic classes, under morphisms of projective algebrai ..."
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Abstract. This paper announces results on the behavior of some important algebraic and topological invariants — Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. — and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas obtained relate global invariants to singularities of general complex algebraic (or analytic) maps. These results, new even for complex manifolds, are applied to obtain a version of GrothendieckRiemannRoch, a calculation of Todd classes of toric varieties, and an explicit formula for the number of integral points in a polytope in Euclidean space with integral vertices. Consider first the behavior of the classical EulerPoincare characteristic e(X) = ∑ (−1) i rankH i (X) i under a (surjective) projective morphism f: X → Y of projective (possibly singular) algebraic varieties. Such a morphism can be stratified with subvarieties as strata. In particular, there is a filtration of Y by closed subvarieties, underlying a Whitney stratification, φ ⊂ Y0 ⊂ · · · ⊂ Ys = Y of strictly increasing dimension, such that Yi −Yi−1 is a union of smooth manifolds of the same dimension and such that the restriction of f to f −1 (Yi − Yi−1) is a locally trivial map of Whitney stratified spaces. (In the results it will suffice to have dim Yi < dim Yi+1 and dim f −1 (x) constant over “strata ” Yi − Yi−1.) We recall the definition of the normal cone CZW of an irreducible subvariety Z of a variety W: n n+1
On the geometry of some strata of unisingular curves
, 2008
"... We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection ..."
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We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection rings of the closures of the strata. The rational equivalence classes of some geometric cycles on the strata are calculated. As an application we give an example when the proper stratum is not affine. As an auxiliary problem we discuss the collision of two singular points, restrictions on possible resulting singularity types and solve the collision problem in several cases. Then we present some cases of enumeration of
LINS NETO’S EXAMPLES OF FOLIATIONS AND THE MORI CONE OF BLOWUPS OF P 2
, 901
"... Abstract. We use a family of algebraic foliations given by A. Lins Neto to provide new evidences to a conjecture, related to the HarbourneHirschowitz’s one and implying the Nagata’s conjecture, which concerns the structure of the Mori cone of blowups of P 2 at very general points. Also, we give an ..."
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Abstract. We use a family of algebraic foliations given by A. Lins Neto to provide new evidences to a conjecture, related to the HarbourneHirschowitz’s one and implying the Nagata’s conjecture, which concerns the structure of the Mori cone of blowups of P 2 at very general points. Also, we give an explicit family of smooth projective rational surfaces X such that the set of faces of the Mori cone NE(X) meeting the region (KX · z = 0) (resp., (KX · z> 0)) is not finite. 1.