scholarly journals A note on k-very ampleness of line bundles on general blow-ups of hyperelliptic surfaces

2015 ◽  
Vol 86 (1) ◽  
pp. 69-77 ◽  
Author(s):  
Łucja Farnik
Keyword(s):  
Line Bundles ◽  
Hyperelliptic Surfaces ◽  
Very Ampleness

scholarly journals Asymptotic Syzygies and Higher Order Embeddings

2020 ◽  
Author(s):  
Daniele Agostini
Keyword(s):  
Hilbert Scheme ◽  
Higher Order ◽  
Partial Answer ◽  
Line Bundles ◽  
Smooth Surfaces ◽  
Hilbert Scheme Of Points ◽  
Very Ampleness

Abstract We show that vanishing of asymptotic $p$-th syzygies implies $p$-very ampleness for line bundles on arbitrary projective schemes. For smooth surfaces, we prove that the converse holds when $p$ is small, by studying the Bridgeland–King–Reid–Haiman correspondence for tautological bundles on the Hilbert scheme of points. This extends the previous results by Ein–Lazarsfeld and Ein–Lazarsfeld–Yang and gives a partial answer to some of their questions. As an application of our results, we show how to use syzygies to bound the irrationality of a variety.

scholarly journals Numerical Criteria for Very Ampleness of Divisors on Projective Bundles Over an Elliptic Curve

1996 ◽  
Vol 48 (6) ◽  
pp. 1121-1137 ◽  
Author(s):  
Alberto Alzati ◽  
Marina Bertolini ◽  
Gian Mario Besana
Keyword(s):  
Elliptic Curve ◽  
Numerical Characterization ◽  
Projective Bundles ◽  
Very Ampleness

AbstractLet D be a divisor on a projectivized bundle over an elliptic curve. Numerical conditions for the very ampleness of D are proved. In some cases a complete numerical characterization is found.

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

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.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

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