scholarly journals aCM vector bundles on projective surfaces of nonnegative Kodaira dimension

2021 ◽  
Author(s):  
Edoardo Ballico ◽  
Sukmoon Huh ◽  
Joan Pons-Llopis
Keyword(s):  
Positive Integer ◽  
Line Bundle ◽  
Vector Bundles ◽  
Ample Line Bundle ◽  
General Setting ◽  
Kodaira Dimension ◽  
Line Bundles ◽  
Wide Range ◽  
Wild Representation Type ◽  
Projective Surfaces

In this paper, we contribute to the construction of families of arithmetically Cohen–Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces [Formula: see text] for [Formula: see text] an ample line bundle. In many cases, we show that for every positive integer [Formula: see text] there exists a family of indecomposable aCM vector bundles of rank [Formula: see text], depending roughly on [Formula: see text] parameters, and in particular they are of wild representation type. We also introduce a general setting to study the complexity of a polarized variety [Formula: see text] with respect to its category of aCM vector bundles. In many cases we construct indecomposable vector bundles on [Formula: see text] which are aCM for all ample line bundles on [Formula: see text].

scholarly journals Deligne pairing and Quillen metric

2014 ◽  
Vol 25 (14) ◽  
pp. 1450122 ◽  
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Positive Integer ◽  
Line Bundle ◽  
Ample Line Bundle ◽  
Hermitian Structure ◽  
Line Bundles ◽  
Canonical Isomorphism ◽  
Surjective Morphism ◽  
Kähler Form ◽  
Hermitian Structures ◽  
Determinant Line Bundle

Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over ℂ. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing 〈L,…,L〉 → S with the determinant line bundle [Formula: see text] (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen–Mumford expansion, J. Differential Geom. 78 (2008) 475–496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles [Formula: see text] and 〈L,…,L〉 carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between 〈L,…,L〉 → S and [Formula: see text] is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91–96] between a Deligne paring and a certain determinant line bundle.

scholarly journals Projective surfaces with K-very ample line bundles of degree ≤ 4K + 4

1994 ◽  
Vol 136 ◽  
pp. 57-79 ◽  
Author(s):  
Edoardo Ballico ◽  
Andrew J. Sommese
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Smooth Surface ◽  
Negative Integer ◽  
Line Bundles ◽  
Restriction Map ◽  
Projective Surface ◽  
Projective Surfaces

A line bundle, L, on a smooth, connected projective surface, S, is defined [7] to be k-very ample for a non-negative integer, k, if given any 0-dimensional sub-scheme with length , it follows that the restriction map is onto. L is 1-very ample (respectively 0-very ample) if and only if L is very ample (respectively spanned at all points by global sections). For a smooth surface, S, embedded in projective space by | L | where L is very ample, L being k-very ample is equivalent to there being no k-secant Pk−1 to S containing ≥ k + 1 points of S.

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 Index conditions and cup-product maps on Abelian varieties

2014 ◽  
Vol 25 (04) ◽  
pp. 1450036 ◽  
Author(s):  
Nathan Grieve
Keyword(s):  
Asymptotic Behavior ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Abelian Varieties ◽  
General Setting ◽  
Index Theorem ◽  
Line Bundles ◽  
One Dimensional ◽  
Cup Product ◽  
Product Maps

We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behavior of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which result in a zero map on a one-dimensional locus. We also prove that the hypothesis of our results can be satisfied, in all possible instances, by a particular class of simple abelian varieties. Finally, we discuss the extent to which Mumford's theta groups are applicable in our more general setting.

scholarly journals CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II

2010 ◽  
Vol 21 (04) ◽  
pp. 497-522 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAINAK PODDAR
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Prime Number ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Line Bundles ◽  
Holomorphic Line Bundle ◽  
Stable Vector Bundles

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let [Formula: see text] denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on [Formula: see text] by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

scholarly journals PSEUDO-EFFECTIVE LINE BUNDLES ON COMPACT KÄHLER MANIFOLDS

2001 ◽  
Vol 12 (06) ◽  
pp. 689-741 ◽  
Author(s):  
JEAN-PIERRE DEMAILLY ◽  
THOMAS PETERNELL ◽  
MICHAEL SCHNEIDER
Keyword(s):  
Line Bundle ◽  
Vector Bundles ◽  
Kähler Manifolds ◽  
Line Bundles ◽  
Kahler Manifolds ◽  
Ideal Sheaf ◽  
Multiplier Ideal Sheaf ◽  
Albanese Map ◽  
Multiplier Ideal ◽  
Hard Lefschetz Theorem

The goal of this work is to pursue the study of pseudo-effective line bundles and vector bundles. Our first result is a generalization of the Hard Lefschetz theorem for cohomology with values in a pseudo-effective line bundle. The Lefschetz map is shown to be surjective when (and, in general, only when) the pseudo-effective line bundle is twisted by its multiplier ideal sheaf. This result has several geometric applications, e.g. to the study of compact Kähler manifolds with pseudo-effective canonical or anti-canonical line bundles. Another concern is to understand pseudo-effectivity in more algebraic terms. In this direction, we introduce the concept of an "almost" nef line bundle, and mean by this that the degree of the bundle is nonnegative on sufficiently generic curves. It can be shown that pseudo-effective line bundles are almost nef, and our hope is that the converse also holds true. This can be checked in some cases, e.g. for the canonical bundle of a projective 3-fold. From this, we derive some geometric properties of the Albanese map of compact Kähler 3-folds.

scholarly journals Moduli spaces of the stable vector bundles over abelian surfaces

1980 ◽  
Vol 77 ◽  
pp. 47-60 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Ample Line Bundle ◽  
Abelian Surfaces ◽  
Singular Variety ◽  
Stable Bundles ◽  
Stable Vector Bundles ◽  
Closed Subvariety

Let X be a projective non-singular variety and H an ample line bundle on X. The moduli space of H-stable vector bundles exists by Maruyama [4]. If X is a curve defined over C, the structure of the moduli space (or its compactification) M(X, d, r) of stable vector bundles of degree d and rank r on X is studied in detail. It is known that the variety M(X, d, r) is irreducible. Let L be a line bundle of degree d and let M(X, L, r) denote the closed subvariety of M(X, d, r) consisting of all the stable bundles E with det E = L.

scholarly journals Moduli spaces of vector bundles over ruled surfaces

1999 ◽  
Vol 154 ◽  
pp. 111-122 ◽  
Author(s):  
Marian Aprodu ◽  
Vasile Brînzănescu
Keyword(s):  
Line Bundle ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Ample Line Bundle ◽  
Sufficient Conditions ◽  
Ruled Surface ◽  
Stable Bundles ◽  
Isomorphism Classes ◽  
Numerical Invariants ◽  
Necessary And Sufficient

AbstractWe study moduli spaces M(c1, c2, d, r) of isomorphism classes of algebraic 2-vector bundles with fixed numerical invariants c1, c2, d, r over a ruled surface. These moduli spaces are independent of any ample line bundle on the surface. The main result gives necessary and sufficient conditions for the non-emptiness of the space M(c1, c2, d, r) and we apply this result to the moduli spaces ML(c1, c2) of stable bundles, where L is an ample line bundle on the ruled surface.

scholarly journals Special Ulrich bundles on regular surfaces with non–negative Kodaira dimension

2021 ◽  
Author(s):  
Gianfranco Casnati
Keyword(s):  
Line Bundle ◽  
Ample Line Bundle ◽  
Kodaira Dimension ◽  
Short List ◽  
Regular Surface ◽  
Elliptic Surfaces ◽  
Low Degree ◽  
Rank 2 ◽  
Polarized Surface ◽  
Negative Kodaira Dimension

AbstractLet S be a regular surface endowed with a very ample line bundle $$\mathcal O_S(h_S)$$ O S ( h S ) . Taking inspiration from a very recent result by D. Faenzi on K3 surfaces, we prove that if $${\mathcal O}_S(h_S)$$ O S ( h S ) satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank 2. As applications, we deal with general embeddings of regular surfaces, pluricanonically embedded regular surfaces and some properly elliptic surfaces of low degree in $$\mathbb {P}^{N}$$ P N . Finally, we also discuss about the size of the families of Ulrich bundles on S and we inspect the existence of special Ulrich bundles on surfaces of low degree.

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