Smoothing of rational m-ropes
In a recent paper, Gallego, González and Purnaprajna showed that rational 3-ropes can be smoothed. We generalise their proof and obtain smoothability of rational $m$-ropes for $m \geq 3$.
math.AG↗
arXiv subjects
Elizabeth Gasparim
Publications and source records attributed to Elizabeth Gasparim.
Local holomorphic Euler characteristic and instanton decay
We study the local holomorphic Euler characteristic $χ(x,\mathcal{F})$ of sheaves near a surface singularity obtained from contracting a line $\ell$ inside a smooth surface $Z$. We prove non-existence of sheaves with certain prescribed numerical invariants. Non-existence of instantons on $Z$ with certain charges follows, and we conclude that $\ell^2$ poses an obstruction to instanton decay. A Macaulay 2 algorithm to compute $χ$ is made available at http://www.maths.ed.ac.uk/~s0571100/Instanton/
math.AG↗
Local moduli of holomorphic bundles
We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.
math.AG↗
The Nekrasov Conjecture for Toric Surfaces
The Nekrasov conjecture predicts a relation between the partition function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on R^4, the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.
math.AG↗
Atiyah Jones conjecture for blown-up surfaces
We show that if the Atiyah Jones conjecture holds for a surface $X,$ then it also holds for the blow-up of $X$ at a point. Since the conjecture is known to hold for ${\mathbb P}^2$ and for ruled surfaces, it follows that the conjecture is true for all rational surfaces.
math.AG↗
Multiplicity of complex hypersurface singularities, Rouche' satellites and Zariski's problem
Soient $f,g\colon (\hbox{åC}^n,0) \to (\hbox{åC},0)$ des germes de fonctions holomorphes réduits. Nous montrons que $f$ et $g$ ont la même multiplicité en 0 si et seulement s'il existe des germes réduits $f'$ et $g'$ analytiquement équivalents à $f$ et $g$, respectivement, tels que $f'$ et $g'$ satisfassent une inégalité du type de Rouché par rapport à un `petit' cercle générique autour de~0. Comme application, nous donnons une reformulation de la question de Zariski sur la multiplicité et une réponse partielle positive à celle--ci.
math.AG↗
Three Applications of Instanton Numbers
We use instanton numbers to: (i) stratify moduli of vector bundles, (ii) calculate relative homology of moduli spaces and (iii) distinguish curve singularities.
math-ph↗
Computing Instanton Numbers of Curve Singularities
We present an algorithm for computing instanton numbers of curve singularities. A comparison is made between these and some other invariants of curve singularities. The algorithm has been implemented in the symbolic computer algebra program Macaulay2, and can be downloaded from http://www.math.nmsu.edu/\~{}iswanson/instanton.m2.}
math.AG↗
Vector bundles on a three dimensional neighborhood of a ruled surface
Let $S$ be a ruled surface inside a smooth threefold $W$ and let $E$ be a vector bundle on a formal neighborhood of $S.$ We find minimal conditions under which the local moduli space of $E$ is finite dimensional and smooth. Moreover, we show that $E$ is a flat limit of a flat family of vector bundles whose general element we describe explicitly.
math.AG↗
Two applications of instanton numbers
The two applications are: 1. sometimes instanton numbers stratify moduli of bundles better than Chern numbers. 2. sometimes instanton numbers distinguish singularities better than the classical numerical invariants.
math.AG↗
Vector Bundles on a Neighborhood of an Exceptional Curve and Elementary Transformations
Let W be the germ of a smooth complex surface around an exceptional curve and let E be a rank 2 vector bundle on W. We study the cohomological properties of a finite sequence $E_i, 1 \leq i \leq t$ of rank 2 vector bundles canonically associated to E. We calculate numerical invariants of E in terms of the $E_i.$ If S is a compact complex smooth surface and E is a rank 2 bundle on the blow- up of S at a point, we show that all values of $c_2(E)-c_2(π_* E^{\vee\vee})$ that are numerically possible are actually attained.
math.AG↗
The Classification of Surfaces
This is an expository paper which presents the holomorphic classification of rational complex surfaces from a simple and intuitive point of view, which is not found in the literature. Our approach is to compare this classification with the topological classification of real surfaces.
math-ph↗
Moduli of bundles on the blown-up plane
Let M(j) denote the moduli space of bundles on the blown-up plane which restrict to the exceptional divisor as O(j)+O(-j). We show that there is a topological embedding of M(j) into M(j+1).
math.AG↗
Chern Classes of Bundles over Rational Surfaces
Consider the blow up $π: \widetilde{X} \to X$ of a rational surface $X$ at a point. Let $\widetilde{V}$ be a holomorphic bundle over $\widetilde{X}$ whose restriction to the exceptional divisor equals ${\cal{O}(j) \oplus {\cal O}(-j)$ and define $V =(π_*\widetilde{V})^{\vee \vee}.$ Friedman and Morgan gave the following bounds for the second Chern classes $j \leq c_2(\widetilde{V}) - c_2(V) \leq j^2.$ We show that these bounds are sharp.
alg-geom↗
On the topology of holomorphic bundles
In this work we study the topology of holomorphic rank two bundles over complex surfaces. We consider bundles that are constructed by glueing and show that under certain conditions the topology of the bundle does not depend on the glueing. We present a simple classification of bundles on blown-up surfaces.
alg-geom↗
Holomorphic bundles on O(-k) are algebraic
We show that holomorphic bundles on O(-k) for k > 0 are algebraic. We also show holomorphic bundles on O(-1) are trivial outside the zero section.
alg-geom↗
Holomorphic Rank Two Vector Bundles on Blow-ups
In this paper we study holomorphic rank two vector bundles on the blow up of $ {\bf C}^2$ at the origin. A classical theorem of Birchoff and Grothendieck says that any holomorphic vector bundle on the projective plane ${\bf P}^1$ splits into a sum of line bundles. If $E$ is a holomorphic vector bundle over the blow up of $ {\bf C}^2$ at the origin, then the restriction of $E$ to the exceptional divisor is a vector bundle over ${\bf P}^1$ and therefore splits. Moreover we assume that $E$ is a rank two bundle that has zero first Chern class. Hence its restriction to the exceptional divisor is of the form $ {\cal O}(j) \oplus {\cal O}(-j) $ for some integer $j.$ We denote by ${\cal M}_j$ the moduli space of equivalence classes (under holomorphic isomorphisms) of rank two holomorphic vector bundles on the blow up of $ {\bf C}^2$ at the origin whose restriction to the exceptional divisor is $ {\cal O}(j) \oplus {\cal O}(-j) .$
alg-geom↗