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 Curves on K3 surfaces in divisibility 2

2021 ◽  
Vol 9 ◽  
Author(s):  
Younghan Bae ◽  
Tim-Henrik Buelles
Keyword(s):  
Modular Forms ◽  
K3 Surfaces ◽  
Initial Condition ◽  
Smooth Curves ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation ◽  
Degeneration Formula ◽  
The Relationship ◽  
Level 2

Abstract We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.

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.

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.

The Varieties of Special Linear Series on a Curve

1985 ◽  
pp. 153-202
Author(s):  
E. Arbarello ◽  
M. Cornalba ◽  
P. A. Griffiths ◽  
J. Harris
Keyword(s):  
Linear Series ◽  
Special Linear

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.

scholarly journals AFFINE GEOMETRY OF STRATA OF DIFFERENTIALS

2017 ◽  
Vol 18 (06) ◽  
pp. 1331-1340 ◽  
Author(s):  
Dawei Chen
Keyword(s):  
Line Bundle ◽  
Algebraic Curve ◽  
Invariant Manifolds ◽  
Affine Variety ◽  
Affine Invariant ◽  
Algebraic Varieties ◽  
Hurwitz Spaces ◽  
Smooth Curves ◽  
Canonical Line Bundle ◽  
Teichmuller Dynamics

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper, we study affine-related properties of strata of $k$ -differentials on smooth curves which parameterize sections of the $k$ th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$ , then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmüller dynamics are affine varieties, and confirm the answer for Teichmüller curves, Hurwitz spaces of torus coverings, hyperelliptic strata as well as some low genus strata.

scholarly journals -algebras associated with curves and rational functions on . I

2014 ◽  
Vol 150 (4) ◽  
pp. 621-667 ◽  
Author(s):  
Robert Fisette ◽  
Alexander Polishchuk
Keyword(s):  
Homotopy Class ◽  
Rational Functions ◽  
Coherent Sheaf ◽  
Linear Series ◽  
Projective Curve ◽  
Canonical Embedding ◽  
Rational Map ◽  
Projective Embedding ◽  
Smooth Projective Curve ◽  
Weierstrass Points

AbstractWe consider the natural$A_{\infty }$-structure on the$\mathrm{Ext}$-algebra$\mathrm{Ext}^*(G,G)$associated with the coherent sheaf$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$on a smooth projective curve$C$, where$p_1,\ldots,p_n\in C$are distinct points. We study the homotopy class of the product$m_3$. Assuming that$h^0(p_1+\cdots +p_n)=1$, we prove that$m_3$is homotopic to zero if and only if$C$is hyperelliptic and the points$p_i$are Weierstrass points. In the latter case we show that$m_4$is not homotopic to zero, provided the genus of$C$is greater than$1$. In the case$n=g$we prove that the$A_{\infty }$-structure is determined uniquely (up to homotopy) by the products$m_i$with$i\le 6$. Also, in this case we study the rational map$\mathcal{M}_{g,g}\to \mathbb{A}^{g^2-2g}$associated with the homotopy class of$m_3$. We prove that for$g\ge 6$it is birational onto its image, while for$g\le 5$it is dominant. We also give an interpretation of this map in terms of tangents to$C$in the canonical embedding and in the projective embedding given by the linear series$|2(p_1+\cdots +p_g)|$.

Syzygies of canonical curves and special linear series

1986 ◽  
Vol 275 (1) ◽  
pp. 105-137 ◽  
Author(s):  
Frank-Olaf Schreyer
Keyword(s):  
Linear Series ◽  
Special Linear ◽  
Canonical Curves

scholarly journals ON TWO CONJECTURES FOR CURVES ON K3 SURFACES

2009 ◽  
Vol 20 (12) ◽  
pp. 1547-1560 ◽  
Author(s):  
ANDREAS LEOPOLD KNUTSEN
Keyword(s):  
Linear System ◽  
Additional Condition ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Smooth Curves ◽  
Natural Extensions ◽  
Complete Linear System

We prove that the gonality among the smooth curves in a complete linear system on a K3 surface is constant except for the Donagi–Morrison example. This was proved by Ciliberto and Pareschi under the additional condition that the linear system is ample. The constancy was originally conjectured by Harris and Mumford. As a consequence we prove that exceptional curves on K3 surfaces satisfy the Eisenbud–Lange–Martens–Schreyer conjecture and explicitly describe such curves. They turn out to be natural extensions of the Eisenbud–Lange–Martens–Schreyer examples of exceptional curves on K3 surfaces.

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