The Severi degree is the degree of the Severi variety parametrizing plane curves of degree d with δ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable q, which are conjecturally equal, for large d. At q = 1, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a combinatorial description ...

We prove that there exists a positive integer νn depending only on n such that for every smooth projective n-fold of general type X defined over C, | mKX | gives a birational rational map from X into a projective space for every m ≥ νn. This theorem gives an affirmative answer to Severi’s conjecture. The key ingredients of the proof are the theory of AZD which was originated by the aurhor and t...

In 1932 F. Severi claimed, with an incorrect proof, that every smooth minimal pro-jective surface S such that the bundle Ω 1 S is generically generated by global sections satisfies the topological inequality 2c 2 1 (S) ≥ c2(S). According to Enriques-Kodaira classification, the above inequality is easily verified when the Kodaira dimension of the surface is ≤ 1, while for surfaces of general typ...

The Severi variety parameterizes plane curves of degree d with δ nodes. Its degree is called the Severi degree. For large enough d, the Severi degrees coincide with the Gromov-Witten invariants of CP. Fomin and Mikhalkin (2009) proved the 1995 conjecture that for fixed δ, Severi degrees are eventually polynomial in d. In this paper, we study the Severi varieties corresponding to a large family ...

For a binary quartic form φ without multiple factors, we classify the quartic K3 surfaces φ(x, y) = φ(z, t) whose Néron-Severi group is (rationally) generated by lines. For generic binary forms φ, ψ of prime degree without multiple factors, we prove that the Néron-Severi group of the surface φ(x, y) = ψ(z, t) is rationally generated by lines.

We classify morphisms from proper varieties to Brauer– Severi varieties, which generalizes the classical correspondence between morphisms to projective space and globally generated invertible sheaves. As an application, we study del Pezzo surfaces of large degree with a view towards Brauer–Severi varieties, and recover classical results on rational points, the Hasse principle, and weak approxim...

Severi developed a theory of correspondences in a series of papers which appeared in 1933, introducing the notions of valences and indices. One of the results achieved by Severi is a formula for the virtual number of fixed points of a correspondence on a smooth projective surface X. These papers are part of Severi’s attempt to develop a theory of the series of equivalences on a surface. In fact...

We apply Tate’s conjecture on algebraic cycles to study the Néron-Severi groups of varieties fibered over a curve. This is inspired by the work of Rosen and Silverman, who carry out such an analysis to derive a formula for the rank of the group of sections of an elliptic surface. For a semistable fibered surface, under Tate’s conjecture we derive a formula for the rank of the group of sections ...