arXiv subjects
Publications and source records attributed to Abdelmoubine Amar Henni.
We study some Huybrechts and Lehn framed sheaves on the Fano 3-fold given by blowing-up the 3-projective space at a point. In contrast with the cases of curves and surfaces, there are very few examples in higher dimensions. In this notes we give a new example of such pairs in dimension 3 and prove that the moduli space under study is fine, quasi-projective and unobstructed.
We consider an extension of the instanton bundles definition, given by Casnati-Coskun-Genk-Malaspina, for Fano threefolds, in order to include non locally-free ones on the blow-up $\widetilde{\mathbb{P}^{3}},$ of the projective $3-$space at a point. With the proposed definition, we prove that any reflexive instanton sheaf must be locally free, and that the strictly torsion free instanton sheaves have singularities of pure dimension $1.$ We construct examples and study their $μ-$stability. Furthermore, these sheaves will play a role in (partially) compactifying the t'Hooft component of the moduli space of instantons, on $\widetilde{\mathbb{P}^{3}}.$ Finally, examples of these are shown to be smooth and smoothable.
We define the analogue of instanton sheaves on the blow-up $\widetilde{\mathbb{P}^n}$ of the $n-$dimensional projective space at a point. We choose appropriate polarisation on $\widetilde{\mathbb{P}^n}$ and construct rank $2$ examples of locally free and non locally free (but torsion free) type. In general, the defined instantons also turn out to be cohomology of monads, although non linear ones. Moreover, in the five dimensional case, we show that there are continuous families of them that fill, at least, a smooth component in the moduli of semi-stable sheaves.
We give an ADHM description of the Quot scheme of points ${\rm Quot}_{\mathbb{C}^{n}}(c,r),$ of length $c$ and rank $r$ on affine spaces $\mathbb{C}^{n}$ which naturally extends both Baranovsky's representation of the punctual Quot scheme on a smooth surface and the Hilbert scheme of points on affine spaces $\mathbb{C}^{n},$ described by the first author and M. Jardim. Using results on the variety of commuting matrices, and combining them with our construction, we prove new properties concerning irreducibility and reducedness of ${\rm Quot}_{\mathbb{C}^{n}}(c,r)$ and its punctual version ${\rm Quot}_{\mathbb{Y}}^{[p]}(c,r),$ where $p$ is a fixed point on a smooth affine variety $\mathbb{Y}.$ In this last case we also study a connectedness result, for some special cases of higher $r$ and $c.$
Let $\mathbb{T}$ be the three dimensional torus acting on $\mathbb{P}^{3}$ and $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c)$ be the fixed locus of the corresponding action on the moduli space of rank $2$ framed instanton sheaves on $\mathbb{P}^{3}.$ In this work, we prove that $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c)$ consist only of non locally-free instanton sheaves whose double dual is the trivial bundle $\mathcal{O}_{\mathbb{P}^{3}}^{\oplus 2}$. Moreover, We relate these instantons to Pandharipande-Thomas stable pairs and give a classification of their support. This allows to compute a lower bound on the number of components of $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c).$
We give a linear algebraic and a monadic descriptions of the Hilbert scheme of points on the affine space of dimension $n$ which naturally extends Nakajima's representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of $c$ points on $(\C^3)$ is irreducible for $c\le 10$.
We present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalizes the one on projective spaces. This is done by generalizing the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterization of these sheaves in the particular case of projective spaces.
We construct monads for framed torsion-free sheaves on blow-ups of the complex projective plane at finitely many distinct points. Using these monads we prove that the moduli space of such sheaves is a smooth algebraic variety. Moreover we construct monads for families of such sheaves parameterized by a noetherian scheme S of finite type. A universal monad on the moduli space is introduced and used to prove that the moduli space is fine.
We consider a particular class of holomorphic vector bundles relevant for supersymmetric string theory, called \emph{omalous}, over nonsingular projective varieties. We use monads to construct examples of such bundles over 3-fold hypersurfaces in $\mathbb{P}^{4}$, complete intersection Calabi-Yau manifolds in $\mathbb{P}^{k}$, blow-ups of $\mathbb{P}^{2}$ at $n$ distinct points, and products $\mathbb{P}^{m}\times\mathbb{P}^{n}$.