The Fundamental Group of the Complement of an Algebraic Curve.
The fundamental groupoid scheme and applications
We define a linear structure on Grothendieck’s arithmetic fundamental group of a scheme defined over a field of characteristic 0. It allows us to link the existence of sections of the Galois group to with the existence of a neutral fiber functor on the category which linearizes it. We apply the construction to affine curves and neutral fiber functors coming from a tangent vector at a rational point at infinity, in order to follow this rational point in the universal covering of the affine...
The Fundamental Group-Scheme of an Abelian Variety.
The Galois group of the tangency problem for plane curves.
The Galois structure of S-units
The Galois-module structure of the space of holomorphic differentials of a curve.
The Gaussian measure on algebraic varieties
We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety is dense in the Hilbert space , where dμ denotes the volume form of M and the Gaussian measure on M.
The generalized Hodge and Bloch conjectures are equivalent for general complete intersections
We prove that Bloch’s conjecture is true for surfaces with obtained as -sets of a section of a very ample vector bundle on a variety with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as on holomorphic -forms of , then it acts as on -cycles of degree of . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general implies the...
The generalized Hodge conjecture for stably nondegenerate abelian varieties
The Generic Torelli Theorem for the Prym Map.
The Genus of Curves in P4 and P5.
The Genus of Curves in P4 verifying certain Flag conditions.
The genus of curves over finite fields with many rational points.
The Genus of Maximal Function Fields over Finite Fields.
The Genus of Space Curves.
The geometric and numerical properties of duality in projective algebraic geometry.
The geometric genus of hypersurface singularities
Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.
The geometric structure of A4, the structure of the Prym map, double solids and ...-divisors.
The geometry of Calogero-Moser systems
We give a geometric construction of the phase space of the elliptic Calogero-Moser system for arbitrary root systems, as a space of Weyl invariant pairs (bundles, Higgs fields) on the -th power of the elliptic curve, where is the rank of the root system. The Poisson structure and the Hamiltonians of the integrable system are given natural constructions. We also exhibit a curious duality between the spectral varieties for the system associated to a root system, and the Lagrangian varieties for...
The geometry of configuration spaces for closed chains in two and three dimensions.