1,721,087 research outputs found
On the Hilbert vector of the Jacobian module of a plane curve
Full text linkWe identify several classes of complex projective plane curves C : f 1⁄4 0, for
which the Hilbert vector of the Jacobian module NðfÞ can be completely determined,
namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having
only rational irreducible components. A result due to Hartshorne, on the cohomology of
some rank 2 vector bundles on P2
, is used to get a sharp lower bound for the initial degree
of the Jacobian module NðfÞ, under a semistability condition
On the duals of smooth projective complex hypersurfaces
Full text linkWe show first that a generic hypersurface of degree in the
complex projective space of dimension has at least
one hyperplane section containing exactly ordinary double
points, alias singularities, in general position, and no other
singularities. Equivalently, the dual hypersurface has at least one
normal crossing singularity of multiplicity . Using this result, we show
that the dual of any smooth hypersurface with has at least a very
singular point , in particular a point of multiplicity .Comment: v3. Some details are added to the proof of the main result. This
version will be published in Publicacions Matem\`atique
Construction of free curves by adding lines to a given curve
Full text linkIn the present note we construct new families of free plane curves starting
from a curve and adding high order inflectional tangent lines of , lines
joining the singularities of the curve , or lines in the tangent cone of
some singularities of . These lines have in common that the intersection
consists of a small number of points. We introduce the notion of a
supersolvable plane curve and conjecture that such curves are always free, as
in the known case of line arrangements. Some evidence for this conjecture is
given as well, both in terms of a general result in the case of quasi
homogeneous singularities and in terms of specific examples. We construct a new
example of maximizing curve in degree 8 and the first and unique known example
of maximizing curve in degree 9. In the final section, we use a stronger
version of a result due to Schenck, Terao and Yoshinaga to construct families
of free conic-line arrangements by adding lines to the conic-line arrangements
of maximal Tjurina number recently classified by V. Beorchia and R. M.
Mir\'o-Roig in arXiv:2303.04665.Comment: Version 4, this is the final version with all the referee's comments.
Accepted for publication in Results in Mathematic
Cohomology of algebraic plane curves
No full textOn décrit dans cette thèse les dimensions des groupes quotients gradués associés à la cohomologie du complémentaire d'une courbe plane par rapport à la filtration de Hodge en fonction de certains invariants géométriques. Le cas des courbes à singularités ordinaires est détaillé. En particulier, on trouve le polynôme de Hodge-Deligne d'une courbe C quelconque à singularités isolées et celui de son complémentaire duquel on déduit les nombres de Hodge mixtes ainsi que les nombres de Betti correspondants. Dans le cas des courbes dont les singularités sont des nœuds et des points triples ordinaires, on donne des relations importantes avec l'algèbre de Milnor du polynôme homogène f qui définit C, les syzygies de l'idéal Jacobien de f et la filtration par l'ordre de pôle du groupe cohomologique d'ordre 2 du complémentaire de la courbe.We describe in this thesis the dimensions of the graded quotients of the cohomology of a plane complement curve with respect to the Hodge filtration in terms of simple geometric invariants. The case of curves with ordinary singularities is discussed in details. In particular, we find the Hodge-Deligne polynomial of any curve C with isolated singularities and that of its complement, from which we can compute the mixed Hodge numbers of the second cohomology group of the complement of the curve, and consequently the correspondant Betti numbers. Furthermore, in the case of curves with ordinary double and triple points, we give relations to the Milnor algebra of the homogeneous polynomial f defining C, to the syzygies of the Jacobian ideal of f and pole order filtration on the second cohomology group of the curve complement
Hyperplane arrangements
No full textCette thèse étudie la fibre de Milnor d'un arrangement d'hyperplans complexe central, et l'opérateur de monodromie sur ses groupes de cohomologie. On s'intéresse à la problématique suivante : peut-on déterminer l'opérateur de monodromie, ou au moins les nombres de Betti de la fibre de Milnor, à partir de l'information contenue dans le treillis d'intersection de l'arrangement? On donne deux théorèmes d'annulation des sous-espaces propres non triviaux de l'opérateur de monodromie. Le premier résultat s'applique à une large classe d'arrangements, le deuxième à des arrangements de droites projectives tels qu'il existe une droite contenant exactement un point de multiplicité supérieure ou égale à trois. Dans le dernier chapitre, on considère la structure de Hodge mixte des groupes de cohomologie de la fibre de Milnor d'un arrangement central et essentiel dans l'espace complexe de dimension quatre. On donne ensuite l'équivalence entre la trivialité de la monodromie, la nullité des coefficients non entiers du spectre de l'arrangement, et la nullité des nombres de Hodge mixtes des groupes de cohomologie de la fibre de Milnor.This Ph.D.thesis studies the Milnor fiber of a central complex hyperplane arrangement, and the monodromy operator on its cohomology groups. Our aim is to study the following open question: is it possible to determinate the monodromy operator, or at least the Betti numbers of the Milnor fiber, just using the information contained in the intersection lattice of the arrangement? We give two vanishing results on the non trivial eigenspaces of the monodromy. The first one applies to a large class of arrangements, and the second one to projective line arrangements with a line containing exactly one point of multiplicity greater or equal to three.Then we consider the mixed Hodge structure of the cohomology groups of the Milnor fiber, for a central and essential hyperplane arrangement in the complex space of dimension four. In this case, we give the equivalence between triviality of the monodromy, Tate properties, and nullity of the non integer spectrum's coefficients.Keywords: hyperplane arrangement, intersection lattice, Milnor fiber, monodromy
On the syzygies and Hodge theory of nodal hypersurfaces
No full textv3. Some applications to the deformation theory of nodal surfaces in are addedWe give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases. A classical result by Severi on the position of the singularities of a nodal surface in is improved and applications to deformation theory of nodal surfaces are given
Higher order Jacobians, Hessians and Milnor algebras
Full text linkWe introduce and study higher order Jacobian ideals, higher order and mixed Hessians, higher order polar maps, and higher order Milnor algebras associated to a reduced projective hypersurface. We relate these higher order objects to some standard graded Artinian Gorenstein algebras, and we study the corresponding Hilbert functions and Lefschetz properties
ADDITION-DELETION RESULTS FOR THE MINIMAL DEGREE OF A JACOBIAN SYZYGY OF A UNION OF TWO CURVES
Full text linkLet C: f=0 be a reduced curve in the complex projective plane. The minimal degree mdr(f) of a Jacobian syzygy for f, which is the same as the minimal degree of a derivation killing f, is an important invariant of the curve C, for instance it can be used to determine whether C is free or nearly free. In this note we study the relations of this invariant with a decomposition of C as a union of two curves and , without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer information on this invariant in this setting. Using this, we give some geometrical criteria, the first ones of this type in the existing literature as far as we know, for a line to be a jumping line for the rank 2 vector bundle of logarithmic vector fields along a reduced curve C
Intersection cohomology of hypersurfaces
Full text linkBekannte Theoreme von Carlson und Griffiths gestatten es, die Variation von Hodgestrukturen assoziiert zu einer Familie von glatten Hyperflächen sowie das Cupprodukt auf der mittleren Kohomologie explizit zu beschreiben. Wir benutzen M. Saitos Theorie der gemischten Hodgemoduln, um diesen Kalkül auf die Variation der Hodgestruktur der Schnittkohomologie von Familien nodaler Hyperflächen zu verallgemeinern.Well known theorems of Carlson and Griffiths provide an explicit description of the variation of Hodge structures associated to a family of smooth hypersurfaces together with the cupproduct pairing on the middle cohomology. We give a generalization to families of nodal hypersurfaces using M. Saitos theory of mixed Hodge modules
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