Higher rank Clifford indices of curves on a K3 surface

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

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CLIFFORD'S THEOREM AND HIGHER RANK VECTOR BUNDLES

International Journal of Mathematics ◽

10.1142/s0129167x02001484 ◽

2002 ◽

Vol 13 (07) ◽

pp. 785-796 ◽

Cited By ~ 16

Author(s):

VINCENT MERCAT

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Smooth Curve ◽

Clifford Index ◽

Precise Bound ◽

Rank 2 ◽

Higher Rank ◽

Rank Vector ◽

Semistable Vector Bundle ◽

Rank 2 Bundles

We give here a refinement of the classical Clifford's theorem for the upper bound of the number of independent global sections of a semistable vector bundle on a smooth curve. We also conjecture a new version of this theorem that takes into account the Clifford index of the curve. In the case of a bi-elliptic curve we obtain a very precise bound. Finally we study the case of rank 2 bundles.

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Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians

Complex Manifolds ◽

10.1515/coma-2015-0007 ◽

2015 ◽

Vol 2 (1) ◽

Author(s):

Svetlana Ermakova

Keyword(s):

Vector Bundle ◽

Complete Intersection ◽

Finite Rank ◽

Vector Bundles ◽

Complete Intersections ◽

Finite Codimension ◽

Line Bundles ◽

Direct Sum ◽

Rank Vector

AbstractIn this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem.We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.

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Higher-rank zeta functions for elliptic curves

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1912023117 ◽

2020 ◽

Vol 117 (9) ◽

pp. 4546-4558 ◽

Cited By ~ 1

Author(s):

Lin Weng ◽

Don Zagier

Keyword(s):

Finite Field ◽

Zeta Function ◽

Elliptic Curves ◽

Riemann Hypothesis ◽

Vector Bundles ◽

Zeta Functions ◽

Smooth Curve ◽

Isomorphism Classes ◽

Higher Rank ◽

Semistable Vector Bundles

In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve X over a finite fieldFqand any integern≥1bywhere the sum is over isomorphism classes ofFq-rational semistable vector bundles V of rank n on X with degree divisible by n. This function, which agrees with the usual Artin zeta function ofX/Fqifn=1, is a rational function ofq−swith denominator(1−q−ns)(1−qn−ns)and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet serieswhere the sum is now over isomorphism classes ofFq-rational semistable vector bundles V of degree 0 on X, is equal to∏k=1∞ζX/Fq(s+k),and use this fact to prove the Riemann hypothesis forζX,n(s)for all n.

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Linear Codes Obtained from Projective and Grassmann Bundles on Curves

Contemporary Mathematics ◽

10.37256/cm.142020449 ◽

2020 ◽

Author(s):

Edoardo Ballico

Keyword(s):

Vector Bundle ◽

Elliptic Curves ◽

Vector Bundles ◽

Linear Codes ◽

Smooth Curve ◽

Set Up ◽

Semistable Vector Bundle ◽

Double Coverings

We use split vector bundles on an arbitrary smooth curve defined over Fq to get linear codes (following the general set-up considered by S. H. Hansen and T. Nakashima), generalizing two quoted results by T. Nakashima. If p ≠ 2 for all integers d, g ≥ 2, r > 0 such that either r is odd or d is even we prove the existence of a smooth curve C of genus g defined over Fq and a p-semistable vector bundle E on C such that rank(E) = r, deg(E) = d and E is defined over Fq. Most results for particular curves are obtained taking double coverings or triple coverings of elliptic curves.

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The Clifford index of line bundles on a 2-elementary K3 surface given by a double cover of a Del Pezzo surface

Geometriae Dedicata ◽

10.1007/s10711-014-9971-5 ◽

2014 ◽

Vol 176 (1) ◽

pp. 329-343 ◽

Cited By ~ 1

Author(s):

Kenta Watanabe

Keyword(s):

Double Cover ◽

Pezzo Surface ◽

Line Bundles ◽

K3 Surface ◽

Del Pezzo Surface ◽

Clifford Index ◽

Del Pezzo

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On geometry of the midlocus associated to a smooth curve in plane and space

Filomat ◽

10.2298/fil1808977a ◽

2018 ◽

Vol 32 (8) ◽

pp. 2977-2990

Author(s):

Azeb Alghanemi ◽

Peter Giblin

Keyword(s):

Plane Curve ◽

Smooth Curve ◽

Space Curve ◽

Regular Space ◽

Geometric Conditions ◽

Special Cases ◽

Smooth Plane ◽

Surface Singularities

The singularities of the midpoint map associated to a smooth plane curve, which is a map from the plane to the plane, are classified. The midlocus associated to a regular space curve is introduced. The geometric conditions for the midlocus of a space curve to have a crosscap or an S?1 singularities are investigated. A more general map, the ?-point map, associated to a space curve is introduced and many known surface singularities are realized as a special cases of this construction.

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Geometry of Vector Bundle Extensions and Applications to a Generalised Theta Divisor

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15233 ◽

2013 ◽

Vol 112 (1) ◽

pp. 61

Author(s):

George H. Hitching

Keyword(s):

Moduli Space ◽

Tangent Cone ◽

Vector Bundles ◽

Smooth Curve ◽

Line Bundles ◽

Singularity Theorem ◽

Theta Divisor ◽

Stable Point ◽

Roch Theorem ◽

Complex Projective

Let $E$ and $F$ be vector bundles over a complex projective smooth curve $X$, and suppose that $0 \to E \to W \to F \to 0$ is a nontrivial extension. Let $G \subseteq F$ be a subbundle and $D$ an effective divisor on $X$. We give a criterion for the subsheaf $G(-D) \subset F$ to lift to $W$, in terms of the geometry of a scroll in the extension space ${\mathbf{P}} H^{1}(X, \mathrm{Hom}(F, E))$. We use this criterion to describe the tangent cone to the generalised theta divisor on the moduli space of semistable bundles of rank $r$ and slope $g-1$ over $X$, at a stable point. This gives a generalisation of a case of the Riemann-Kempf singularity theorem for line bundles over $X$. In the same vein, we generalise the geometric Riemann-Roch theorem to vector bundles of slope $g-1$ and arbitrary rank.

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Connected Numbers and the Embedded Topology of Plane Curves

Canadian Mathematical Bulletin ◽

10.4153/cmb-2017-066-5 ◽

2018 ◽

Vol 61 (3) ◽

pp. 650-658 ◽

Cited By ~ 3

Author(s):

Taketo Shirane

Keyword(s):

Plane Curve ◽

Smooth Curve ◽

Plane Curves ◽

Irreducible Curve ◽

Galois Cover ◽

Splitting Number

AbstractThe splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

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