Related papers: Normal Bundle of Rational Curves and Waring Decomp…

We study vector bundles on curves with rational tails and their smoothings and give a sufficient condition for the general fibre to be balanced.

We study the stable pairs theory of local curves in 3-folds with descendent insertions. The rationality of the partition function of descendent invariants is established for the full local curve geometry (equivariant with respect to the…

We attempt to describe the rank 2 vector bundles on a curve C which are specializations of the trivial bundle. We get a complete classifications when C is Brill-Noether generic, or when it is hyperelliptic; in both cases all limit vector…

In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)-automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and…

Let X be a compact Kaehler manifold. We expect that any direct sum decomposition of the tangent bundle T(X) comes from a splitting of the universal covering space of X as a product of manifolds, in such a way that the given decomposition of…

We determine all triples $(e,d,n)$ for which a general degree $d$ hypersurface $X\subset \mathbb{P}^n$ contains a degree $e$ rational curve $C$ with balanced restricted tangent bundle $T_X|_C$. In addition, we show how to compute explicit…

In this article we present a study of embeddings of complex supermanifolds. We are broadly guided by the question: when will a submanifold of a split supermanifold itself be split? As an application of our study, we will address this…

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed…

Graph partitioning, or the dividing of a graph into two or more parts based on certain conditions, arises naturally throughout discrete mathematics, and problems of this kind have been studied extensively. In the 1990s, Ando conjectured…

This paper deals with rational curves and birational contractions on irreducible holomorphically symplectic manifold. We survey some recent results about minimal rational curves, their deformations, extremal rays associated with these…

Let $C$ be a smooth, projective, geometrically irreducible curve defined over $\mathbb{R}$ such that $C(\mathbb{R}) = \emptyset$. Let $r>0$ and $d$ be integers which are coprime. Let $L$ be a line bundle on $C$ which corresponds to an…

For a semistable reflexive sheaf $E$ of rank $r$ and $c_1=a$ on $\P^n$ and an integer $d$ such that $r|ad$, we give sufficient conditions so that the restriction of $E$ on a generic rational curve of degree $d$ is balanced, i.e. a twist of…

Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition…

We study the stability of the normal bundle of canonical genus $8$ curves and prove that on a general curve the bundle is stable. The proof rests on Mukai's description of these curves as linear sections of a Grassmannian $\mathrm{G}(2,6)$.…

The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The…

Any arrangement of hyperplanes in general position in $P^n$ can be regarded as a divisor with normal crossing. We study the bundles of logarithmic 1-forms corresponding to such divisors` from the point of view of classification of vector…

Recently L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (whose link $M$ is a rational homology sphere) with the Seiberg-Witten invariant of $M$…

We study the derived category of coherent sheaves on various versions of moduli space of vector bundles on curves by the Borel-Weil-Bott theory for loop groups and $\Theta$-stratification, and construct a semiorthogonal decomposition with…

We describe a sequence of smooth quotients of the Deligne-Mumford moduli space ${\mathbb R}\overline{\mathcal M}_{0,\ell+1}$ of real rational curves with $\ell\!+\!1$ conjugate pairs of marked points that terminates at ${\mathbb…

For a smooth projective curve C with genus g >= 2 and a degree 1 line bundle L on C, let M := SU_{C}(r;L) be the moduli space of stable vector bundles of rank r and with the fixed determinant L. In this paper, we study the small rational…