scholarly journals Non-defectivity of Grassmannian bundles over a curve

2016 ◽  
Vol 27 (07) ◽  
pp. 1640002 ◽  
Author(s):  
Insong Choe ◽  
George H. Hitching
Keyword(s):  
Vector Bundles ◽  
Manuscripta Math ◽  
Maximal Degree ◽  
Geometric Proof ◽  
Secant Varieties ◽  
Grassmann Bundle ◽  
Positive Rank

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

scholarly journals A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve

2014 ◽  
Vol 25 (05) ◽  
pp. 1450047 ◽  
Author(s):  
Insong Choe ◽  
G. H. Hitching
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Upper Bound ◽  
Topological Structure ◽  
Vector Bundles ◽  
Geometric Interpretation ◽  
Maximal Degree ◽  
Secant Varieties ◽  
Extension Spaces ◽  
Orthogonal Case

A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on t(V), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.

scholarly journals On the Rationality of Certain Strata of the Lange Stratification of Stable Vector Bundles on Curves

2001 ◽  
Vol 8 (4) ◽  
pp. 665-668
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundles ◽  
Maximal Degree ◽  
Projective Curve ◽  
Fiber Structure ◽  
Generic Fiber ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Natural Way

Abstract Let 𝑋 be a smooth projective curve of genus 𝑔 ≥ 2 and 𝑆(𝑟, 𝑑) the moduli scheme of all rank 𝑟 stable vector bundles of degree 𝑑 on 𝑋. Fix an integer 𝑘 with 0 < 𝑘 < 𝑟. H. Lange introduced a natural stratification of 𝑆(𝑟, 𝑑) using the degree of a rank 𝑘 subbundle of any 𝐸 ∈ 𝑆(𝑟, 𝑑) with maximal degree. Every non-dense stratum, say 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎), has in a natural way a fiber structure ℎ : 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎) → Pic𝑎(𝑋) × Pic𝑏(𝑋) with ℎ dominant. Here we study the rationality or the unirationality of the generic fiber of ℎ.

Maximal Subbundles of Rank 2 Vector Bundles on Projective Curves

2000 ◽  
Vol 43 (2) ◽  
pp. 129-137 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Stable Rank ◽  
Maximal Degree ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Projective Curves ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

AbstractLet E be a stable rank 2 vector bundle on a smooth projective curve X and V(E) be the set of all rank 1 subbundles of E with maximal degree. Here we study the varieties (non-emptyness, irreducibility and dimension) of all rank 2 stable vector bundles, E, on X with fixed deg(E) and deg(L), L ∈ V(E) and such that .

scholarly journals Secant varieties and Hirschowitz bound on vector bundles over a curve

2010 ◽  
Vol 133 (3-4) ◽  
pp. 465-477 ◽  
Author(s):  
Insong Choe ◽  
George H. Hitching
Keyword(s):  
Vector Bundles ◽  
Secant Varieties

scholarly journals Quantitative properties of the non-properness set of a polynomial map, a positive characteristic case

2019 ◽  
Vol 19 (10) ◽  
pp. 2050192
Author(s):  
Zbigniew Jelonek ◽  
Michał Lasoń
Keyword(s):  
Upper Bound ◽  
Positive Characteristic ◽  
Manuscripta Math ◽  
Geometric Proof ◽  
Real Numbers ◽  
Affine Spaces ◽  
Polynomial Curves ◽  
Polynomial Map ◽  
Characteristic Case ◽  
Quantitative Properties

Let [Formula: see text] be a generically finite polynomial map of degree [Formula: see text] between affine spaces. In [Z. Jelonek and M. Lasoń, Quantitative properties of the non-properness set of a polynomial map, Manuscripta Math. 156(3–4) (2018) 383–397] we proved that if [Formula: see text] is the field of complex or real numbers, then the set [Formula: see text] of points at which [Formula: see text] is not proper is covered by polynomial curves of degree at most [Formula: see text]. In this paper, we generalize this result to positive characteristic. We provide a geometric proof of an upper bound by [Formula: see text].

scholarly journals Stable vector bundles on curves: numerical invariants of their extremal subbundles

2001 ◽  
Vol 89 (1) ◽  
pp. 46
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Maximal Degree ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Numerical Invariants

Let $X$ be a smooth projective curve of genus $g \geq 2$ and $E$ a rank $r$ vector bundle on $X$. If $1 \leq t < r$ set $s_t(E)$:= sup $\lbrace t$(deg($E$)) - $r$(deg($F$)), where $F$ is a rank $t$ subsheaf of $E \rbrace $. Here we construct rank $r$ stable vector bundles $E$ on $X$ such that the sequence $ \lbrace s_t(E) \rbrace _{1 \leq t<r}$ has a prescribed value and the set of all subsheaves of $E$ with maximal degree may be explicitely described.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals Fano 3-folds from homogeneous vector bundles over Grassmannians

2021 ◽  
Author(s):  
Lorenzo De Biase ◽  
Enrico Fatighenti ◽  
Fabio Tanturri
Keyword(s):  
Vector Bundles ◽  
Homogeneous Vector

AbstractWe rework the Mori–Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.

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