Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denot... more Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by K X. Let M H denote the moduli space of semistable Higgs Gp(2n, C)-bundles over X of fixed topological type. The complex variety M H has a natural holomorphic symplectic structure. On the other hand, for any ≥ 1, the Liouville symplectic from on the total space of K X defines a holomorphic symplectic structure on the Hilbert scheme Hilb (K X) parametrizing the zero-dimensional subschemes of K X. We relate the symplectic form on Hilb (K X) with the symplectic form on M H .
International Journal of Geometric Methods in Modern Physics, 2010
Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denot... more Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by KX. Let [Formula: see text] denote the moduli space of semistable Higgs Gp (2n, ℂ)-bundles over X of fixed topological type. The complex variety [Formula: see text] has a natural holomorphic symplectic structure. On the other hand, for any ℓ ≥ 1, the Liouville symplectic from on the total space of KX defines a holomorphic symplectic structure on the Hilbert scheme Hilb ℓ(KX) parametrizing the zero-dimensional subschemes of KX. We relate the symplectic form on Hilb ℓ(KX) with the symplectic form on [Formula: see text].
Let $X$ be a smooth projective algebraic surface of Picard rank one with very ample canonical bun... more Let $X$ be a smooth projective algebraic surface of Picard rank one with very ample canonical bundle $K_X$. We further assume that $q -1 \le χ(\mathcal{O}_X$. In this article, we will study the existence of the Ulrich bundle and its stability property of it with respect to $K_X$.
Let S ⊂ℙ^3 be a very general sextic surface over complex numbers. Let ℳ(H, c_2) be the moduli spa... more Let S ⊂ℙ^3 be a very general sextic surface over complex numbers. Let ℳ(H, c_2) be the moduli space of rank 2 stable bundles on S with fixed first Chern class H and second Chern class c_2. In this article we study the configuration of points of certain reduced zero dimensional subschemes on S satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general memeber of the moduli space for small c_2. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components of ℳ(H, 11) and prove the conjecture partially. We will also show that ℳ(H, c_2) is irreducible for c_2 ≤ 10 .
Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. ... more Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle $E_{C, A}$ associated to a line bundle $A$ ion $C$ such that $|A|$ is a pencil on $C$ and computes the Clifford index of $C$. We give a necessary and sufficient condition for $E_{C, A}$ being semistable.
Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will prove that... more Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will prove that under certain condition the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system $|L|$.
Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will give an al... more Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will give an alternative and elementary proof of Lelli Chiesa's Theorem in the case of $r= 2$. More precisely we will prove that that under certain condition the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system $|L|$. Using Lelli Chiesa's theorem for $r \ge 3$ we also extend Lelli Chiesa's Theorem in the case of $r= 2$ in weaker condition.
Let (X,H) be a polarized smooth projective algebraic surface and E is globally generated, stable ... more Let (X,H) be a polarized smooth projective algebraic surface and E is globally generated, stable vector bundle on X . Then the Syzygy bundle ME associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of ME with respect to H .
Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. Let $\mathcal{... more Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. Let $\mathcal{M}(H, c_2)$ be the moduli space of rank $2$ stable bundles on $S$ with fixed first Chern class $H$ and second Chern class $c_2$. In this article we will classify the obstructed bundles in $ \mathcal{M}(H, c_2)$ for small $c_2$. Using this classification we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components of $\mathcal{M}(H, 11)$ and prove the conjecture partially. We will also show that $\mathcal{M}(H, c_2)$ is irreducible for $c_2 \le 10
Let $S$ be a very general smooth hypersurface of degree $6$ in $\mathbb{P}^3$. In this paper we w... more Let $S$ be a very general smooth hypersurface of degree $6$ in $\mathbb{P}^3$. In this paper we will prove that the moduli space of $\mu$-stable rank $2$ torsion free sheaves with respect to hyperplane section having $c_1 = \mathcal{O}_S(1)$, with fixed $c_2 \ge 27$ is irreducible.
Let X be a smooth projective complex curve of genus g ≥ 2 and let M X (2,Λ) be the moduli space o... more Let X be a smooth projective complex curve of genus g ≥ 2 and let M X (2,Λ) be the moduli space of semi-stable rank-2 vector bundles over X with fixed determinant Λ. We show that the wobbly locus, i.e. the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field, is a union of divisors 𝓦 k ⊂ M X (2,Λ). We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors 𝓦 k in the Picard group of M X (2, Λ).
Let X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this ... more Let X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this short note we will show that the moduli space of rank z stable vector bundles with determinant isomorphic to Lx , where Lx denotes the line bundle corresponding to a point x ∊ X, is isomorphic to a certain variety of lines in the moduli space of S-equivalence classes of semistable bundles of rank 2 with trivial determinant.
Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denot... more Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by K X. Let M H denote the moduli space of semistable Higgs Gp(2n, C)-bundles over X of fixed topological type. The complex variety M H has a natural holomorphic symplectic structure. On the other hand, for any ≥ 1, the Liouville symplectic from on the total space of K X defines a holomorphic symplectic structure on the Hilbert scheme Hilb (K X) parametrizing the zero-dimensional subschemes of K X. We relate the symplectic form on Hilb (K X) with the symplectic form on M H .
Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denot... more Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by K X. Let M H denote the moduli space of semistable Higgs Gp(2n, C)-bundles over X of fixed topological type. The complex variety M H has a natural holomorphic symplectic structure. On the other hand, for any ≥ 1, the Liouville symplectic from on the total space of K X defines a holomorphic symplectic structure on the Hilbert scheme Hilb (K X) parametrizing the zero-dimensional subschemes of K X. We relate the symplectic form on Hilb (K X) with the symplectic form on M H .
International Journal of Geometric Methods in Modern Physics, 2010
Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denot... more Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by KX. Let [Formula: see text] denote the moduli space of semistable Higgs Gp (2n, ℂ)-bundles over X of fixed topological type. The complex variety [Formula: see text] has a natural holomorphic symplectic structure. On the other hand, for any ℓ ≥ 1, the Liouville symplectic from on the total space of KX defines a holomorphic symplectic structure on the Hilbert scheme Hilb ℓ(KX) parametrizing the zero-dimensional subschemes of KX. We relate the symplectic form on Hilb ℓ(KX) with the symplectic form on [Formula: see text].
Let $X$ be a smooth projective algebraic surface of Picard rank one with very ample canonical bun... more Let $X$ be a smooth projective algebraic surface of Picard rank one with very ample canonical bundle $K_X$. We further assume that $q -1 \le χ(\mathcal{O}_X$. In this article, we will study the existence of the Ulrich bundle and its stability property of it with respect to $K_X$.
Let S ⊂ℙ^3 be a very general sextic surface over complex numbers. Let ℳ(H, c_2) be the moduli spa... more Let S ⊂ℙ^3 be a very general sextic surface over complex numbers. Let ℳ(H, c_2) be the moduli space of rank 2 stable bundles on S with fixed first Chern class H and second Chern class c_2. In this article we study the configuration of points of certain reduced zero dimensional subschemes on S satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general memeber of the moduli space for small c_2. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components of ℳ(H, 11) and prove the conjecture partially. We will also show that ℳ(H, c_2) is irreducible for c_2 ≤ 10 .
Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. ... more Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle $E_{C, A}$ associated to a line bundle $A$ ion $C$ such that $|A|$ is a pencil on $C$ and computes the Clifford index of $C$. We give a necessary and sufficient condition for $E_{C, A}$ being semistable.
Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will prove that... more Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will prove that under certain condition the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system $|L|$.
Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will give an al... more Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will give an alternative and elementary proof of Lelli Chiesa's Theorem in the case of $r= 2$. More precisely we will prove that that under certain condition the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system $|L|$. Using Lelli Chiesa's theorem for $r \ge 3$ we also extend Lelli Chiesa's Theorem in the case of $r= 2$ in weaker condition.
Let (X,H) be a polarized smooth projective algebraic surface and E is globally generated, stable ... more Let (X,H) be a polarized smooth projective algebraic surface and E is globally generated, stable vector bundle on X . Then the Syzygy bundle ME associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of ME with respect to H .
Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. Let $\mathcal{... more Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. Let $\mathcal{M}(H, c_2)$ be the moduli space of rank $2$ stable bundles on $S$ with fixed first Chern class $H$ and second Chern class $c_2$. In this article we will classify the obstructed bundles in $ \mathcal{M}(H, c_2)$ for small $c_2$. Using this classification we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components of $\mathcal{M}(H, 11)$ and prove the conjecture partially. We will also show that $\mathcal{M}(H, c_2)$ is irreducible for $c_2 \le 10
Let $S$ be a very general smooth hypersurface of degree $6$ in $\mathbb{P}^3$. In this paper we w... more Let $S$ be a very general smooth hypersurface of degree $6$ in $\mathbb{P}^3$. In this paper we will prove that the moduli space of $\mu$-stable rank $2$ torsion free sheaves with respect to hyperplane section having $c_1 = \mathcal{O}_S(1)$, with fixed $c_2 \ge 27$ is irreducible.
Let X be a smooth projective complex curve of genus g ≥ 2 and let M X (2,Λ) be the moduli space o... more Let X be a smooth projective complex curve of genus g ≥ 2 and let M X (2,Λ) be the moduli space of semi-stable rank-2 vector bundles over X with fixed determinant Λ. We show that the wobbly locus, i.e. the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field, is a union of divisors 𝓦 k ⊂ M X (2,Λ). We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors 𝓦 k in the Picard group of M X (2, Λ).
Let X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this ... more Let X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this short note we will show that the moduli space of rank z stable vector bundles with determinant isomorphic to Lx , where Lx denotes the line bundle corresponding to a point x ∊ X, is isomorphic to a certain variety of lines in the moduli space of S-equivalence classes of semistable bundles of rank 2 with trivial determinant.
Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denot... more Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by K X. Let M H denote the moduli space of semistable Higgs Gp(2n, C)-bundles over X of fixed topological type. The complex variety M H has a natural holomorphic symplectic structure. On the other hand, for any ≥ 1, the Liouville symplectic from on the total space of K X defines a holomorphic symplectic structure on the Hilbert scheme Hilb (K X) parametrizing the zero-dimensional subschemes of K X. We relate the symplectic form on Hilb (K X) with the symplectic form on M H .
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