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.

scholarly journals Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

2020 ◽  
Vol 2020 (21) ◽  
pp. 8139-8182 ◽  
Author(s):  
Jarosław Buczyński ◽  
Nathan Ilten ◽  
Emanuele Ventura
Keyword(s):  
Linear System ◽  
Smooth Curve ◽  
Rational Curves ◽  
Arithmetic Genus ◽  
Smooth Curves ◽  
Low Degree ◽  
Complete Linear System ◽  
Osculating Spaces ◽  
Special Case ◽  
Schubert Cycles

Abstract In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree $d$ rational curves in $\mathbb{P}^n$ when $d-n\leq 3$ and $d<2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d<2n$, the arithmetic genus of any non-degenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d-n$.

scholarly journals Pencils and nets on curves arising from rank $1$ sheaves on K3 surfaces

2018 ◽  
Vol 122 (2) ◽  
pp. 197
Author(s):  
Nils Henry Rasmussen
Keyword(s):  
Linear System ◽  
Simple Proof ◽  
Final Section ◽  
Smooth Curve ◽  
Base Point ◽  
K3 Surfaces ◽  
Small Degree ◽  
K3 Surface ◽  
Free Sheaf ◽  
Torsion Free

Let $S$ be a K3 surface, $C$ a smooth curve on $S$ with $\mathcal{O} _S(C)$ ample, and $A$ a base-point free $g^2_d$ on $C$ of small degree. We use Lazarsfeld-Mukai bundles to prove that $A$ is cut out by the global sections of a rank $1$ torsion-free sheaf $\mathcal{G} $ on $S$. Furthermore, we show that $c_1(\mathcal{G} )$ with one exception is adapted to $\mathcal{O} _S(C)$ and satisfies $\operatorname{Cliff} (c_1(\mathcal{G} )_{|C})\leq \operatorname{Cliff} (A)$, thereby confirming a conjecture posed by Donagi and Morrison. We also show that the same methods can be used to give a simple proof of the conjecture in the $g^1_d$ case.In the final section, we give an example of the mentioned exception where $h^0(C,c_1(\mathcal{G} )_{|C})$ is dependent on the curve $C$ in its linear system, thereby failing to be adapted to $\mathcal{O} _S(C)$.

scholarly journals Smooth Curves on Projective $K3$ Surfaces

2002 ◽  
Vol 90 (2) ◽  
pp. 215 ◽  
Author(s):  
Andreas Leopold Knutsen
Keyword(s):  
Sufficient Conditions ◽  
K3 Surfaces ◽  
Picard Group ◽  
Necessary And Sufficient Conditions ◽  
K3 Surface ◽  
Minimal Rank ◽  
Irreducible Curve ◽  
Smooth Curves ◽  
Intersection Of Quadrics ◽  
Necessary And Sufficient

In this paper we give for all $n \geq 2$, $d>0$, $g \geq 0$ necessary and sufficient conditions for the existence of a pair $(X,C)$, where $X$ is a $K3$ surface of degree $2n$ in $\mathrm{P}^{n+1}$ and $C$ is a smooth (reduced and irreducible) curve of degree $d$ and genus $g$ on $X$. The surfaces constructed have Picard group of minimal rank possible (being either $1$ or $2$), and in each case we specify a set of generators. For $n \geq 4$ we also determine when $X$ can be chosen to be an intersection of quadrics (in all other cases $X$ has to be an intersection of both quadrics and cubics). Finally, we give necessary and sufficient conditions for $\mathcal O_C (k)$ to be non-special, for any integer $k \geq 1$.

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.

scholarly journals Curves with Two Pencils and Associated Maps to Projective Spaces

2006 ◽  
Vol 13 (3) ◽  
pp. 411-417
Author(s):  
Edoardo Ballico
Keyword(s):  
Linear System ◽  
Projective Spaces ◽  
Free Resolution ◽  
Homogeneous Ideal ◽  
Projective Curve ◽  
Minimal Free Resolution ◽  
Complete Linear System

Abstract Let 𝑋 be a smooth and connected projective curve. Assume the existence of spanned 𝐿 ∈ Pic𝑎(𝑋), 𝑅 ∈ Pic𝑏(𝑋) such that ℎ0(𝑋, 𝐿) = ℎ0(𝑋, 𝑅) = 2 and the induced map ϕ 𝐿,𝑅 : 𝑋 → 𝐏1 × 𝐏1 is birational onto its image. Here we study the following question. What can be said about the morphisms β : 𝑋 → 𝐏𝑅 induced by a complete linear system |𝐿⊗𝑢⊗𝑅⊗𝑣| for some positive 𝑢, 𝑣? We study the homogeneous ideal and the minimal free resolution of the curve β(𝑋).

scholarly journals Autoequivalences of twisted K3 surfaces

2019 ◽  
Vol 155 (5) ◽  
pp. 912-937 ◽  
Author(s):  
Emanuel Reinecke
Keyword(s):  
K3 Surfaces ◽  
K3 Surface ◽  
Lattice Structures ◽  
Hodge Structures ◽  
Derived Equivalences

Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index $1$or $2$in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.

Isogenies Between K3 Surfaces Over

2020 ◽  
Author(s):  
Ziquan Yang
Keyword(s):  
Abelian Varieties ◽  
K3 Surfaces ◽  
Main Step ◽  
Shimura Varieties ◽  
K3 Surface ◽  
Perfect Field ◽  
Finite Height ◽  
Mild Assumption

Abstract We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over ${\mathbb{C}}$ to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over $\bar{{\mathbb{F}}}_p$ by prescribing linear algebraic data when $p$ is large. The main step is to show that isogenies between Kuga–Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on $p$.

ON THE DEGREE AND THE BIRATIONALITY OF THE SECOND ADJUNCTION MAPPING

1999 ◽  
Vol 10 (06) ◽  
pp. 707-719 ◽  
Author(s):  
MAURO C. BELTRAMETTI ◽  
ANDREW J. SOMMESE
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Mild Condition ◽  
Three Dimensional ◽  
Ample Line Bundle ◽  
Kodaira Dimension ◽  
Projective Manifold ◽  
Very Ample Line Bundle ◽  
System 2 ◽  
Complete Linear System

Let ℒ be a very ample line bundle on ℳ, a projective manifold of dimension n ≥3. Under the assumption that Kℳ + (n-2) ℒ has Kodaira dimension n, we study the degree of the map ϕ associated to the complete linear system |2(KM + (n-2) L)|, where (M, L) is the first reduction of (ℳ, ℒ). In particular we show that under a number of conditions, e.g. n ≥ 5 or Kℳ + (n-3)ℒ having nonnegative Kodaira dimension, the degree of ϕ is one, i.e. ϕ is birational. We also show that under a mild condition on the linear system |KM + (n-2) L| satisfied for all known examples, ϕ is birational unless (ℳ, ℒ) is a three dimensional variety with very restricted invariants. Moreover there is an example with these invariants such that deg ϕ= 2.

scholarly journals Ordinary reduction of K3 surfaces

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Fedor Bogomolov ◽  
Yuri Zarhin
Keyword(s):  
Number Field ◽  
Field Extension ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Zero Set ◽  
Algebraic Field ◽  
Density Zero ◽  
Archimedean Place ◽  
Ordinary Reduction

AbstractLet X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

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