Remarks on the Algebraic Approach to Intersection Theory.
Remarks on the generalized index of an analytic improper intersection
This article continues the investigation of the analytic intersection algorithm from the perspective of deformation to the normal cone, carried out in the previous papers of the author [7, 8, 9]. The main theorem asserts that, given an analytic set V and a linear subspace S, every collection of hyperplanes, admissible with respect to an algebraic bicone B, realizes the generalized intersection index of V and S. This result is important because the conditions for a collection of hyperplanes to be...
Residue currents and complexity problems
Residues of regular and meromorphic differential forms.
Semigroups associated to singular points of plane curves.
Simple conditions on conic varieties.
Skew-Symmetric Vanishing Lattices and Their Monodromy Groups.
Skew-Symmetric Vanishing Lattices and Their Monodromy Groups. II.
Solutions to the XXX type Bethe ansatz equations and flag varieties
We consider a version of the A N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl N+1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B N and...
Subvarieties of Segre varieties and trisecant lines.
Supplement to the paper "Improper intersections in complex analytic geometry" (Dissertationes Math. 391 (2001))
Sur l'abaissement de la classe d'une courbe produit par la présence d'un point de rebroussement
Sur le théorème du produit
On donne des versions raffinées effectives du théorème du produit de G. Faltings et de son principal corollaire. Le théorème montre que si l’ensemble des zéros d’indice d’un polynôme multihomogène a une composante commune avec l’ensemble des zéros d’indice alors cette composante, sous-variété d’un produit d’espaces projectifs, est elle-même un produit à condition que les rapports des degrés de soient grands en fonction de . Le corollaire le plus utile implique que, sous une condition plus...
Sur les orbites d’un sous-groupe sphérique dans la variété des drapeaux
Soient un groupe algébrique complexe réductif et connexe, un sous-groupe de Borel de et un sous-groupe sphérique de . Soit un plongement -équivariant de . Nous savons que n’a qu’un nombre fini d’orbites dans ; nous montrons qu’il n’en a qu’un nombre fini dans . Soit l’adhérence dans d’une orbite de dans et l’adhérence d’une orbite de dans . Si est toroïdal, nous montrons que l’intersection est propre dans et la décrivons ensemblistement. Si de plus est lisse,...
The border cases of the lifting theorem for surfaces in P4.
The Chern character for Lie-Rinehart algebras
Let be a commutative -algebra where is a ring containing the rationals. We prove the existence of a Chern character for Lie-Rinehart algebras over A with values in the Lie-Rinehart cohomology of L which is independent of choice of a -connection. Our result generalizes the classical Chern character from the -theory of to the algebraic De Rham cohomology.
The Chow-Witt ring.
The cone of curves of a K3 surface.
The degrees of certain strata of the dual variety
The Euler number of the normalization of an algebraic threefold with ordinary singularities
By a classical formula due to Enriques, the Euler number χ(X) of the non-singular normalization X of an algebraic surface S with ordinary singularities in P³(ℂ) is given by χ(X) = n(n²-4n+6) - (3n-8)m + 3t - 2γ, where n is the degree of S, m the degree of the double curve (singular locus) of S, t is the cardinal number of the triple points of S, and γ the cardinal number of the cuspidal points of S. In this article we shall give a similar formula for an algebraic threefold with ordinary singularities...