The canonical ring of a 3-connected curve (original) (raw)

On the canonical ring of curves and surfaces

2011

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring R(C, ω C ) = k≥0 H 0 (C, ω C ⊗k ) is generated in degree 1 if C is numerically 4-connected, not hyperelliptic and even (i.e. with ω C of even degree on every component).

On Clifford’s theorem for singular curves

Let C be a 2-connected projective curve either reduced with planar singularities or contained in a smooth algebraic surface and let S be a subcanonical cluster (i.e. a 0-dimensional scheme such that the space H 0 (C, I S K C ) contains a generically invertible section). Under some general assumptions on S or C we show that h 0 (C, I S K C ) ≤ p a (C) − 1 2 deg(S) and if equality holds then either S is trivial, or C is honestly hyperelliptic or 3-disconnected.

The degree of the generators of the canonical ring of surfaces of general type with p g = 0

Archiv der Mathematik, 1997

Upper bounds for the degree of the generators of the canonical rings of surfaces of general type were found by Ciliberto [C]. In particular it was established that the canonical ring of a minimal surface of general type with p g = 0 is generated by its elements of degree lesser or equal to 6, ([C], th. (3.6)). This was the best bound possible to obtain at the time, since Reider's results, [R], were not yet available. In this note, this bound is improved in some cases (theorems (3.1), (3.2)). In particular it is shown that if K 2 ≥ 5, or if K 2 ≥ 2 and |2K S | is base point free this bound can be lowered to 4. This result is proved by showing first that, under the same hypothesis, the degree of the bicanonical map is lesser or equal to 4 if K 2 ≥ 3, (theorem (2.1)), implying that the hyperplane sections of the bicanonical image have not arithmetic genus 0. The result on the generation of the canonical ring then follows by the techniques utilized in [C]. Notation and conventions. We will denote by S a projective algebraic surface over the complex field. Usually S will be smooth, minimal, of general type. We denote by K S , or simply by K if there is no possibility of confusion, a canonical divisor on S. As usual, for any sheaf F on S, we denote by h i (S, F) the dimension of the cohomology space H i (S, F), and by p g and q the geometric genus and the irregularity of S. By a curve on S we mean an effective, non zero divisor on S. We will denote the intersection number of the divisors C, D on S by C • D and by C 2 the self-intersection of the divisor C. We denote by ≡ the linear equivalence for divisors on S. |D| will be the complete linear system of the effective divisors D ′ ≡ D, and φ D : S → P(H 0 (S, O S (D) ∨) = |D| ∨ the natural rational map defined by |D|. We will denote by Σ d the rational ruled surface P(O P 1 ⊕ O P 1 (d)), for d ≥ 0. ∆ ∞ will denote the section of Σ d with minimum self-intersection −d and Γ will be a fibre of the projection to P 1 .

A Note on 1-Forms for Reduced Curve Singularities

Cornell University - arXiv, 2017

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity, in particular of a complete intersection, and use them to compare the Milnor number, the Tjurina number and the dimension of the torsion part of the 1-forms. 1 Classical results This note was motivated by a recent preprint of A. Dimca [Di17] one 1forms of irreducible plane curve singularities. We generalize his results to not necessarily irreducible complete intersection curve singularities by deducing them from classical results (partly in German), which are somewhat scattered in the literature and apparently not well known. We collect them here with reference to the original sources. Consider a reduced complex curve singularity (X, 0) ⊂ (C n , 0) defined by an ideal I ⊂ O C n ,0 with r = r(X, 0) branches. Let n : (X,0) → X, 0) be the normalization, where (X,0) is the multi-germ consisting of r smooth branches. We set

Ordinary Singularities of Algebraic Curves

Canadian mathematical bulletin, 1981

Let A be the local ring at a singular point p of an algebraic reduced curve. Let M (resp. M l5 ..., M h) be the maximal ideal of A (resp. of Â). In this paper we want to classify ordinary singularities p with reduced tangent cone: Spec(G(A)). We prove that G(A) is reduced if and only if: p is an ordinary singularity, and the vector spaces Hom(M"IM? +1 t k) span the vector space Hom(M n /M n+1 , k). If the points of the projectivized tangent cone Proj(G(A)) are in generic position then p is an ordinary singularity if and only if G(A) is reduced. We give an example which shows that the preceding equivalence is not true in general.

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.

The canonical model of a singular curve

Geometriae Dedicata, 2009

We give refined statements and modern proofs of Rosenlicht's results about the canonical model C ′ of an arbitrary complete integral curve C. Notably, we prove that C and C ′ are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C ′ is equal to the blowup of C with respect to the canonical sheaf ω. We also prove some new results: we determine just when C ′ is rational normal, arithmetically normal, projectively normal, and linearly normal.