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Register a new userStability of some vector bundles on Hilbert schemes of points on K3 surfaces
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Let $X$ be a projective K3 surfaces. In two examples where there exists a fine moduli space $M$ of stable vector bundles on $X$, isomorphic to a Hilbert scheme of points, we prove that the universal family $mathcal{E}$ on $Xtimes M$ can be understood as a complete flat family of stable vector bundles on $M$ parametrized by $X$, which identifies $X$ with a smooth connected component of some moduli space of stable sheaves on $M$.
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We show that any polarized K3 surface supports special Ulrich bundles of rank 2.
Over a smooth and proper complex scheme, the differential Galois group of an integrable connection may be obtained as the closure of the transcendental monodromy representation. In this paper, we employ a completely algebraic variation of this idea by restricting attention to connections on trivial vector bundles and replacing the fundamental group by a certain Lie algebra constructed from the regular forms. In more detail, we show that the differential Galois group is a certain ``closure of the aforementioned Lie algebra. This is then applied to construct connections on curves with prescribed differential Galois group.
Let $K$ be a discretely-valued field. Let $Xrightarrow Spec K$ be a surface with trivial canonical bundle. In this paper we construct a weak Neron model of the schemes $Hilb^n(X)$ over the ring of integers $Rsubseteq K$. We exploit this construction in order to compute the Motivic Zeta Function of $Hilb^n(X)$ in terms of $Z_X$. We determine the poles of $Z_{Hilb^n(X)}$ and study its monodromy property, showing that if the monodromy conjecture holds for $X$ then it holds for $Hilb^n(X)$ too. Sit $K$ corpus cum absoluto ualore discreto. Sit $ Xrightarrow Spec K$ leuigata superficies cum canonico fasce congruenti $mathcal{O}_X$. In hoc scripto defecta Neroniensia paradigmata $Hilb^n(X)$ schematum super annulo integrorum in $K$ corpo, $R subset K$, constituimus. Ex hoc, Functionem Zetam Motiuicam $Z_{Hilb^n(X)}$, dato $Z_X$, computamus. Suos polos statuimus et suam monodromicam proprietatem studemus, coniectura monodromica, quae super $X$ ualet, ualere super $Hilb^n(X)$ quoque demostrando.
In [DKO] we constructed virtual fundamental classes $[[ Hilb^m_V ]]$ for Hilbert schemes of divisors of topological type m on a surface V, and used these classes to define the Poincare invariant of V: (P^+_V,P^-_V): H^2(V,Z) --> Lambda^* H^1(V,Z) x Lambda^* H^1(V,Z) We conjecture that this invariant coincides with the full Seiberg-Witten invariant computed with respect to the canonical orientation data. In this note we prove that the existence of an integral curve $C subset V$ induces relations between some of these virtual fundamental classes $[[Hilb^m_V ]]$. The corresponding relations for the Poincare invariant can be considered as algebraic analoga of the fundamental relations obtained in [OS].
The Debarre-Voisin hyperkahler fourfolds are built from alternating $3$-forms on a $10$-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate, in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general $1$-parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3 surface.
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