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.

scholarly journals A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES

2015 ◽  
Vol 16 (4) ◽  
pp. 859-877 ◽  
Author(s):  
Benjamin Bakker
Keyword(s):  
Hilbert Scheme ◽  
Rational Curve ◽  
K3 Surface ◽  
Higher Dimensional ◽  
Extremal Curve ◽  
Symplectic Varieties

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^{2}=-2$. We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $M$ deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface, an extremal curve class $R\in H_{2}(M,\mathbb{Z})$ in the Mori cone is the line in a Lagrangian $n$-plane $\mathbb{P}^{n}\subset M$ if and only if certain intersection-theoretic criteria are met. In particular, any such class satisfies $(R,R)=-\frac{n+3}{2}$, and the primitive such classes are all contained in a single monodromy orbit.

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 SINGULAR CURVES ON A K3 SURFACE AND LINEAR SERIES ON THEIR NORMALIZATIONS

2007 ◽  
Vol 18 (06) ◽  
pp. 671-693 ◽  
Author(s):  
FLAMINIO FLAMINI ◽  
ANDREAS LEOPOLD KNUTSEN ◽  
GIANLUCA PACIENZA
Keyword(s):  
Linear Systems ◽  
Picard Group ◽  
Linear Series ◽  
K3 Surface ◽  
Nodal Curves

We study the Brill–Noether theory of the normalizations of singular, irreducible curves on a K3 surface. We introduce a singular Brill–Noether number ρsing and show that if Pic (K3) = ℤ[L], there are no [Formula: see text]'s on the normalizations of irreducible curves in |L|, provided that ρsing < 0. We give examples showing the sharpness of this result. We then focus on the case of hyperelliptic normalizations, and classify linear systems |L| containing irreducible nodal curves with hyperelliptic normalizations, for ρsing < 0, without any assumption on the Picard group.

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 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.

scholarly journals Birational classification of curves on rational surfaces

2010 ◽  
Vol 199 ◽  
pp. 43-93
Author(s):  
Alberto Calabri ◽  
Ciro Ciliberto
Keyword(s):  
Linear System ◽  
Plane Curve ◽  
Rational Surface ◽  
Minimal Degree ◽  
Classification Theorem ◽  
Plane Curves ◽  
Cremona Transformation ◽  
Rational Surfaces

AbstractIn this paper we consider the birational classification of pairs (S, ℒ), withSa rational surface andℒa linear system onS. We give a classification theorem for such pairs, and we determine, for each irreducible plane curveB, itsCremona minimalmodels, that is, those plane curves which are equivalent toBvia a Cremona transformation and have minimal degree under this condition.

scholarly journals Divisor Varieties of Symmetric Products

2021 ◽  
Author(s):  
John Sheridan
Keyword(s):  
Algebraic Curves ◽  
Linear Series ◽  
Symmetric Product ◽  
Fixed Degree ◽  
Symmetric Products ◽  
Projective Varieties ◽  
General Curve ◽  
Higher Dimensional ◽  
Detailed Picture

Abstract The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and dimension) are smooth, irreducible projective varieties of known dimension. For higher dimensional varieties, the story is less well understood. Our purpose in this paper is to study in detail one class of higher dimensional examples where one can hope for a quite detailed picture, namely (the spaces parametrizing) divisors on the symmetric product of a curve.

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