Surfaces of General Type (original) (raw)

On the genus of reducible surfaces and degenerations of surfaces

Annales de l’institut Fourier, 2007

In this paper we deal with a reducible projective surface X with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the ω-genus p ω (X) of X, i.e. the dimension of the vector space of global sections of the dualizing sheaf ω X . Then we prove that, when X is smoothable, i.e. when X is the central fibre of a flat family π : X → ∆ parametrized by a disc, with smooth general fibre, then the ω-genus of the fibres of π is constant.

The base components of the dualizing sheaf of a curve on a surface

Archiv der Mathematik, 2008

This note studies the structure of the divisorial fixed part of |ω D | for a 1-connected curve D on a smooth surface S. It is shown that if the divisorial fixed part F of |ω D | is non empty then it has arithmetic genus ≤ 0 and each component of F is a smooth rational curve. The stucture of curves D, with non empty divisorial fixed part F for |ω D |, is also described.

Topics in deformation and moduli theory for singularities on curves and surfaces

The aim of this thesis is to contribute to the understanding of moduli of isolated singularities in dimension one and two. Historically, Riemann classified the possible conformal structures on a compact Riemann surface. In algebraic geometry the problem of moduli has gotten increasing attention. Local moduli of singularities is one aspect, and Zariski considered in [Zar73] this problem for plane curve singularities of the form x m + y m+1 . Later Laudal and Pfister took a systematic approach to this problem for plane curve singularities of quasihomogeneous type, see .

On the canonical degrees of curves in varieties of general type

Geometric and Functional Analysis, 2012

In this paper, we work in the framework of complex analytic varieties; without contrary mention, varieties are assumed to be irreducible (and reduced). If C is a projective curve, we let g C be its geometric genus (namely, the genus of its desingularization) and χ(C) = 2 − 2g C its geometric Euler characteristic; we also write deg C L for the degree of a line bundle L on C.

Roitman?s theorem for singular complex projective surfaces

Duke Mathematical Journal, 1996

Let X be a complex projective surface with arbitrary singularities. We construct a generalized Abel-Jacobi map A 0 (X) → J 2 (X) and show that it is an isomorphism on torsion subgroups. Here A 0 (X) is the appropriate Chow group of smooth 0-cycles of degree 0 on X, and J 2 (X) is the intermediate Jacobian associated with the mixed Hodge structure on H 3 (X). Our result generalizes a theorem of Roitman for smooth surfaces: if X is smooth then the torsion in the usual Chow group A 0 (X) is isomorphic to the torsion in the usual Albanese variety J 2 (X) ∼ = Alb(X) by the classical Abel-Jacobi map. * Members of GNSAGA of CNR, partially supported by ECC Science Plan * * Partially supported by NSF grants If X is a smooth projective surface over the complex numbers C, the classical Abel-Jacobi map goes from the Chow group A 0 (X) of cycles of degree 0 to the (group underlying the) Albanese Variety Alb(X). Roitman's Theorem states that this map induces an isomorphism on torsion subgroups. (See [9] for a nice compendium).