On the projective normality of complete linear series on an algebraic curve

1986 ◽  
Vol 83 (1) ◽  
pp. 73-90 ◽  
Author(s):  
Mark Green ◽  
Robert Lazarsfeld
Keyword(s):  
Algebraic Curve ◽  
Linear Series ◽  
Projective Normality ◽  
Complete Linear Series

scholarly journals The extremal secant conjecture for curves of arbitrary gonality

2017 ◽  
Vol 153 (2) ◽  
pp. 347-357
Author(s):  
Michael Kemeny
Keyword(s):  
Algebraic Curve ◽  
Linear Series ◽  
Line Bundles ◽  
Projective Normality ◽  
Complete Linear Series ◽  
Invent Math

We prove the Green–Lazarsfeld secant conjecture [Green and Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), 73–90; Conjecture (3.4)] for extremal line bundles on curves of arbitrary gonality, subject to explicit genericity assumptions.

scholarly journals K3 carpets on minimal rational surfaces and their smoothings

2021 ◽  
pp. 2150032
Author(s):  
Purnaprajna Bangere ◽  
Jayan Mukherjee ◽  
Debaditya Raychaudhury
Keyword(s):  
Linear Series ◽  
Minimal Degree ◽  
K3 Surface ◽  
Rational Surfaces ◽  
Double Structure ◽  
Higher Dimensional ◽  
Complete Linear Series

In this paper, we study K3 double structures on minimal rational surfaces [Formula: see text]. The results show there are infinitely many non-split abstract K3 double structures on [Formula: see text] parametrized by [Formula: see text], countably many of which are projective. For [Formula: see text] there exists a unique non-split abstract K3 double structure which is non-projective (see [J.-M. Drézet, Primitive multiple schemes, preprint (2020), arXiv:2004.04921 , to appear in Eur. J. Math.]). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless [Formula: see text] is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on [Formula: see text]. Moreover, we show any embedded projective K3 carpet on [Formula: see text] with [Formula: see text] arises as a flat limit of embeddings degenerating to 2:1 morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on [Formula: see text], embedded by a complete linear series are smooth points if and only if [Formula: see text]. In contrast, Hilbert points corresponding to projective (split) K3 carpets supported on [Formula: see text] and embedded by a complete linear series are always smooth. The results in [P. Bangere, F. J. Gallego and M. González, Deformations of hyperelliptic and generalized hyperelliptic polarized varieties, preprint (2020), arXiv:2005.00342 ] show that there are no higher dimensional analogues of the results in this paper.

On the Proof of a Lemma Enunciated by Severi

1936 ◽  
Vol 32 (2) ◽  
pp. 253-259 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Algebraic Curve ◽  
Algebraic Surface ◽  
Linear Series

1. There is a lemma given by Severi which is of importance because it is used by him in his proof that the number of finite Picard integrals belonging to an algebraic surface is equal to the irregularity of the surface; it is also used by Castelnuovo † in his proof of the same result. The lemma is: If upon an algebraic curve there is an irreducible algebraic series, ∞1, of sets of s points, of index r; and if the sets of sr points which consist of all the r sets which contain a given point (this point taken r times) move in a linear series as this point varies, then any one of the r sets of s points separately moves in a linear series. The proof given by Severi was held satisfactory by Castelnuovot, but I find difficulty in stating it with precision; and Castelnuovo gives an entirely different proof, founded on an enumerative formula due to Schubert (loc. cit. p. 341); this is also the course adopted by Enriques-Chisini.

Some enumerative formulae for algebraic curves

1958 ◽  
Vol 54 (4) ◽  
pp. 399-416 ◽  
Author(s):  
I. G. Macdonald
Keyword(s):  
Equivalence Relation ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Linear Series ◽  
Set Of Points ◽  
Algebraic Correspondence

This paper is in two parts. In Part I we are concerned with one or more linear series on an algebraic curve; we consider a set of points on the curve which are contained with assigned multiplicities in a set of each of the linear series and, by persistent use of Severi's equivalence relation for the united points of an algebraic correspondence with valency, we derive formulae for the number of such sets of points when the constants involved are such as to make this number finite. All this is essentially a generalization of the formula for the number of points in the Jacobian set of a linear series of freedom 1, and the main result is Theorem 3.

Some Results on Surfaces of General Type

2005 ◽  
Vol 57 (4) ◽  
pp. 724-749 ◽  
Author(s):  
B. P. Purnaprajna
Keyword(s):  
Linear Systems ◽  
General Type ◽  
Base Point ◽  
Linear Series ◽  
Ample Divisor ◽  
Surfaces Of General Type ◽  
Projective Normality

AbstractIn this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series. Some of the corollaries of more general results include: results on property Np associated to KS ⊗B⊗n where B is base-point free and ample divisor with B⊗K* nef, results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are optimal.

scholarly journals Green’s conjecture for curves on arbitrary K3 surfaces

2011 ◽  
Vol 147 (3) ◽  
pp. 839-851 ◽  
Author(s):  
Marian Aprodu ◽  
Gavril Farkas
Keyword(s):  
Algebraic Curve ◽  
Complete Solution ◽  
K3 Surfaces ◽  
Linear Series ◽  
Canonical Embedding ◽  
Smooth Curves ◽  
Special Linear

AbstractGreen’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K3 surfaces.

scholarly journals A specialization inequality for tropical complexes

2021 ◽  
Vol 157 (5) ◽  
pp. 1051-1078
Author(s):  
Dustin Cartwright
Keyword(s):  
Arbitrary Dimension ◽  
Linear Series ◽  
Complete Linear Series

We prove a specialization inequality relating the dimension of the complete linear series on a variety to the tropical complex of a regular semistable degeneration. Our result extends Baker's specialization inequality to arbitrary dimension.

On postulation of curves: Embeddings by complete linear series

1984 ◽  
Vol 43 (3) ◽  
pp. 244-249 ◽  
Author(s):  
Edoardo Ballico ◽  
Philippe Ellia
Keyword(s):  
Linear Series ◽  
Complete Linear Series

scholarly journals A Generalisation of a Formula due to Schubert

1958 ◽  
Vol 11 (2) ◽  
pp. 79-82
Author(s):  
J. G. Brennan
Keyword(s):  
Fixed Points ◽  
Algebraic Curve ◽  
Linear Series ◽  
Double Points ◽  
Image Position

Let there be given, on an algebraic curve C, of genus p, a linear series and an algebraic series of index v, both without fixed points. The number of groups of r + 1 points which are common to a set of and a set of has been shown by Schubert (1) to bewhere d is the number of double points of .

scholarly journals Dimension Theory and Degenerations of de Jonquières Divisors

2019 ◽  
Author(s):  
Mara Ungureanu
Keyword(s):  
Linear Series ◽  
General Linear ◽  
Dimension Theory ◽  
Projective Curve ◽  
Expected Number ◽  
Smooth Projective Curve ◽  
Nodal Curves ◽  
General Curve ◽  
Complete Linear Series

Abstract This paper aims at settling the issue of the validity of the de Jonquières formulas. Consider the space of divisors with prescribed multiplicity, or de Jonquières divisors, contained in a linear series on a smooth projective curve. Under the assumption that this space is zero dimensional, the de Jonquières formulas compute the expected number of de Jonquières divisors. Using degenerations to nodal curves we show that, for a general curve equipped with a complete linear series, the space is of expected dimension, which shows that the counts are in fact true. This implies that in the case of negative expected dimension a general linear series on a general curve does not admit de Jonquières divisors of the expected type.

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