In the current paper we prove that any Severi variety on a Hirzebruch surface contains a unique component parameterizing irreducible nodal curves of the given genus in characteristic zero.

We give an exposition of a theorem of Hirzebruch, Kodaira and Yau which proves the uniqueness of the Kähler structure of complex projective space, and of Yau’s resolution of the Severi Conjecture.

We give a short, elementary, and characteristic independent proof of the criterion for motivic isomorphism of two projective quadrics discovered by A. Vishik [24]. We also give a criterion for motivic isomorphism of two Severi-Brauer varieties.

In this note, I review an aspect of some new techniques introduced recently in collaboration with Miguel \'Angel Barja and Rita Pardini: the construction continuous rank function. give a sketch how to use function prove Barja-Clifford-Pardini-Severi inequalities for varieties maximal Albanese dimension obtain classification satisfying equalities.

0. Introduction (a) Statement of the problem and of the main theorem; some references 233 (b) Corollaries of the main theorem 236 (c) Role of the Brill-Noether matrix 238 (d) Heuristic reasoning for the dimension count; the CastelnuovoSeveri-Kleiman conjecture 240 (e) Notations and terminology 242 1. Reduction of the dimension count to the conjecture (a) Heuristic discussion 244 (b) Geometry of...

Soit K un corps infini et A une K-algèbre de dimension finie n. Soit 0 < r < n un entier. Le résultat principal de cet article est le suivant. Sous certaines hypothèses sur A (satisfaites si A est étale), la grassmannienne G(r, A) est K-birationnelle, de manière PGL1(A)-équivariante, au produit de G(pgcd(r, n), A) par un espace affine (théorème 3.3). On en déduit par torsion plusieurs résultats...

The Neron-Severi group of divisor classes modulo algebraic equivalence on a smooth algebraic surface is often not difficult to calculate, and has classically been studied as one of the fundamental invariants of the surface. A more difficult problem is the determination of those divisor classes which can be represented by effective divisors; these divisor classes form a monoid contained in the N...