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Register a new userNormalization of congruence of bitangents to a hypersurface in $mathbb P^3$
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A congruence is a surface in the Grassmannian ${rm Gr}(2, 4)$. In this paper, we consider the normalization of congruence of bitangents to a hypersurface in $mathbb P^3$. We call it the Fano congruence of bitangents. We give a criterion for smoothness of the Fano congruence of bitangents and describe explicitly their degenerations in a general Lefschetz pencil in the space of hypersurfaces in $mathbb P^3$.
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For a reduced hypersurface $V(f) subseteq mathbb{P}^n$ of degree $d$, the Castelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understood when $V(f)$ is smooth, as well as when $V(f)$ has isolated singularities. We study the regularity of $M(f)$ when $V(f)$ has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by $(d-2)(n+1)$, which is the degree of the Hessian polynomial of $f$. However, this is not always the case, and we prove that in $mathbb{P}^n$ the regularity of the Milnor algebra can grow quadratically in $d$.
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We study the spectrum of rank $2$ torsion free sheaves on $mathbb{P}^3$ with aim of producing examples of distinct irreducible components of the moduli space with the same spetrcum answering the question presented by Rao for the case of torsion free sheaves. In order to do so, we provide a full description of the spectrum of the sheaves in the moduli space of semistable rank $2$ torsion free sheaves on $mathbb{P}^3$ with Chern classes $(c_1, c_2,c_3)$ equals to $(-1,2,0)$ and $(0,3,0)$.
We describe new components of the Gieseker--Maruyama moduli scheme $mathcal{M}(n)$ of semistable rank 2 sheaves $E$ on $mathbb{P}^3$ with $c_1(E)=0$, $c_2(E)=n$ and $c_3(E)=0$ whose generic point corresponds to non locally free sheaves. We show that such components grow in number as $n$ grows, and discuss how they intersect the instanton component. As an application, we prove that $mathcal{M}(2)$ is connected, and identify a connected subscheme of $mathcal{M}(3)$ consisting of 7 irreducible components.
We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M(e,n)$ of rank 2 stable vector bundles with the first Chern class $e=0$ or -1 and all possible values of the second Chern class $n$ on the projective 3-space. The generalized null correlation bundles constituting open dense subsets of these components are defined as cohomology bundles of monads whose members are direct sums of line bundles of degrees depending on nonnegative integers $a,b,c$, where $bge a$ and $c>a+b$. We show that, in the wide range when $c>2a+b-e, b>a, (e,a) e(0,0)$, the Ein components are rational, and in the remaining cases they are at least stably rational. As a consequence, the union of the spaces $M(e,n)$ over all $nge1$ contains an infinite series of rational components for both $e=0$ and $e=-1$. Explicit constructions of rationality of Ein components under the above conditions on $e,a,b,c$ and, respectively, of their stable rationality in the remaining cases, are given. In the case of rationality, we construct universal families of generalized null correlation bundles over certain open subsets of Ein components showing that these subsets are fine moduli spaces. As a by-product of our construction, for $c_1=0$ and $n$ even, they provide, perhaps the first known, examples of fine moduli spaces not satisfying the condition $n$ is odd, which is a usual sufficient condition for fineness.
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