scholarly journals Universal polynomials for singular curves on surfaces

2014 ◽  
Vol 150 (7) ◽  
pp. 1169-1182 ◽  
Author(s):  
Jun Li ◽  
Yu-jong Tzeng
Keyword(s):  
Vector Bundle ◽  
Linear System ◽  
Line Bundle ◽  
Vector Bundles ◽  
Projective Surface ◽  
Analytic Singularity ◽  
Smooth Projective Surface ◽  
Chern Numbers ◽  
Cutting Out ◽  
Curves On Surfaces

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a complex smooth projective surface and $L$ be a line bundle on $S$. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system $|L|$ with prescribed singularities is a universal polynomial of Chern numbers of $L$ and $S$, assuming $L$ is sufficiently ample. More generally, we show for vector bundles of any rank and smooth varieties of any dimension, similar universal polynomials also exist and equal the number of singular subvarieties cutting out by sections of the vector bundle. This work is a generalization of Göttsche’s conjecture.

scholarly journals THE DI FRANCESCO–ITZYKSON–GÖTTSCHE CONJECTURES FOR NODE POLYNOMIALS OF ℙ2

2012 ◽  
Vol 23 (04) ◽  
pp. 1250049 ◽  
Author(s):  
NIKOLAY QVILLER
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Projective Surface ◽  
Universal Functions ◽  
Recursive Procedure ◽  
Bell Polynomial ◽  
Nodal Curves ◽  
Chern Numbers ◽  
Classical Shape ◽  
Linear Polynomials

For a smooth, irreducible projective surface S over ℂ, the number of r-nodal curves in an ample linear system [Formula: see text] (where [Formula: see text] is a line bundle on S) can be expressed using the rth Bell polynomial Pr in universal functions ai, 1 ≤ i ≤ r, of (S, [Formula: see text]), which are ℤ-linear polynomials in the four Chern numbers of S and [Formula: see text]. We use this result to establish a proof of the classical shape conjectures of Di Francesco–Itzykson and Göttsche governing node polynomials in the case of ℙ2. We also give a recursive procedure which provides the [Formula: see text]-term of the polynomials ai.

scholarly journals Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

2019 ◽  
Vol 163 (3-4) ◽  
pp. 361-373
Author(s):  
Roberto Laface ◽  
Piotr Pokora
Keyword(s):  
Line Bundle ◽  
Characteristic Zero ◽  
Algebraic Surface ◽  
Intersection Number ◽  
Algebraic Surfaces ◽  
The Self ◽  
Projective Surface ◽  
Intersection Numbers ◽  
Weighted Version ◽  
Smooth Projective Surface

AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

scholarly journals On vector bundles on algebraic surfaces and 0-cycles

1993 ◽  
Vol 130 ◽  
pp. 19-23 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Algebraic Structure ◽  
Vector Bundles ◽  
Algebraic Surfaces ◽  
Projective Surface ◽  
Classical Theorem ◽  
Chern Classes ◽  
Complex Projective Surface ◽  
Holomorphic Structure ◽  
Rank 2

Let X be an algebraic complex projective surface equipped with the euclidean topology and E a rank 2 topological vector bundle on X. It is a classical theorem of Wu ([Wu]) that E is uniquely determined by its topological Chern classes . Viceversa, again a classical theorem of Wu ([Wu]) states that every pair (a, b) ∈ (H (X, Z), Z) arises as topological Chern classes of a rank 2 topological vector bundle. For these results the existence of an algebraic structure on X was not important; for instance it would have been sufficient to have on X a holomorphic structure. In [Sch] it was proved that for algebraic X any such topological vector bundle on X has a holomorphic structure (or, equivalently by GAGA an algebraic structure) if its determinant line bundle has a holomorphic structure. It came as a surprise when Elencwajg and Forster ([EF]) showed that sometimes this was not true if we do not assume that X has an algebraic structure but only a holomorphic one (even for some two dimensional tori (see also [BL], [BF], or [T])).

THE BOGOMOLOV–MIYAOKA–YAU INEQUALITY FOR LOG CANONICAL SURFACES

2001 ◽  
Vol 64 (2) ◽  
pp. 327-343 ◽  
Author(s):  
ADRIAN LANGER
Keyword(s):  
Vector Bundles ◽  
Universal Cover ◽  
Kodaira Dimension ◽  
Projective Surface ◽  
Chern Classes ◽  
Singular Surfaces ◽  
Smooth Projective Surface ◽  
Log Canonical ◽  
Surface Singularities ◽  
Negative Kodaira Dimension

Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov [1, Theorem 5] proved that c21 [les ] 4c2. This was improved to c21 [les ] 3c2 by Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality c21 [les ] 3c2 is attained, for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only possibility). Further generalizations of inequalities for Chern classes for some singular surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6], Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12], Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].In [8] we introduced Chern classes of reflexive sheaves, using Wahl's local Chern classes of vector bundles on resolutions of surface singularities. Here we apply them to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.

Sums of stably trivial vector bundles

1980 ◽  
Vol 87 (1) ◽  
pp. 97-107 ◽  
Author(s):  
Jacques Allard
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Vector Bundles ◽  
Real Vector

We say that a real vector bundle ξ over a finite C.W. complex X is stably trivial of type (n, k) or, simply, of type (n, k) if ξ ⊕ kε ≅ nε, where ε denotes a trivial line bundle. The following theorem is an immediate corollary (see (12)) of a theorem of T. Y. Lam ((9), theorem 2).

scholarly journals ASYMPTOTIC TORSION AND TOEPLITZ OPERATORS

2015 ◽  
Vol 16 (2) ◽  
pp. 223-349 ◽  
Author(s):  
Jean-Michel Bismut ◽  
Xiaonan Ma ◽  
Weiping Zhang
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Differential Form ◽  
Toeplitz Operators ◽  
Vector Bundles ◽  
Direct Image ◽  
Analytic Torsion ◽  
Leading Term ◽  
Positive Line Bundle ◽  
Positive Line

We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles $F_{p}$. For $p\in \mathbf{N}$, the flat vector bundle $F_{p}$ is the direct image of $L^{p}$, where $L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

scholarly journals On the Connectedness of the Real Part of Moduli Spaces of Vector Bundles on Real Algebraic Surfaces

2000 ◽  
Vol 68 (1) ◽  
pp. 41-54
Author(s):  
Edoardo Ballico
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Algebraic Surfaces ◽  
Stable Rank ◽  
Projective Surface ◽  
Chern Classes ◽  
The Real ◽  
Smooth Projective Surface ◽  
Real Algebraic Surfaces

AbstractLet X be a smooth projective surface with q(X) = 0 defined over R and M(X;r;c1;c2;H) the moduli space of H-stable rank r vector bundles on X with Chern classes c1 and c2. Assume either r = 3 and X(R) connected or r = 3 and X(R) =ø or r=2 and X(R) = ø. We prove that quite often M is connected.

scholarly journals Global generation of adjoint bundles

1996 ◽  
Vol 142 ◽  
pp. 5-16 ◽  
Author(s):  
Hajime Tsuji
Keyword(s):  
Line Bundle ◽  
Ample Line Bundle ◽  
Base Point ◽  
Projective Surface ◽  
Smooth Projective Surface

In 1988, I. Reider proved that for a smooth projective surface X and an ample line bundle L on X, Kx + 3L is globally generated and Kx + 4L is very ample ([12]). In fact his theorem is much stronger than this (see [12] for detail). Recently a lot of results have been obtained about effective base point freeness (cf. [1, 3, 8, 13, 14, 15]). In particular J. P. Demailly proved that 2KX + 12nnL is very ample for a smooth projective n-fold X and an ample line bundle L on X. [2] will give a good overview for these recent results. The motivation of these works is the following conjecture posed by T. Fujita.

scholarly journals Stability of tautological bundles on the Hilbert scheme of two points on a surface

2014 ◽  
Vol 214 ◽  
pp. 79-94 ◽  
Author(s):  
Malte Wandel
Keyword(s):  
Vector Bundle ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Stable Vector Bundle ◽  
Free Sheaf ◽  
Smooth Projective Surface ◽  
Rank 2 ◽  
Torsion Free

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

scholarly journals Stable vector bundles on algebraic surfaces II

1973 ◽  
Vol 52 ◽  
pp. 173-195 ◽  
Author(s):  
Fumio Takemoto
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Vector Bundles ◽  
Ample Line Bundle ◽  
Algebraic Surfaces ◽  
Complex Numbers ◽  
Stable Vector Bundle ◽  
Projective Varieties ◽  
Stable Vector Bundles ◽  
Direct Sum

This paper is a continuation of “Stable vector bundles on algebraic surfaces” [10]. For simplicity we deal with non-singular projective varieties over the field of complex numbers. Let W be a variety whose fundamental group is solvable, let H be an ample line bundle on W, and let f: V → W be an unramified covering. Then we show in section 1 that if E is an f*H-stable vector bundle on V then f*E is a direct sum of H-stable vector bundles. In particular f*L is a direct sum of simple vector bundles if L is a line bundle on V.

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