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Register a new userModuli of surfaces in $mathbb{P}^3$
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Authors
Kristin DeVleming
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The goal of this paper is to construct a compactification of the moduli space of degree $d ge 5$ surfaces in $mathbb{P}^3$, i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in $mathbb{P}^3$ and whose boundary points correspond to degenerations of such surfaces. We study a more general problem and consider a divisor $D$ on a Fano variety $Z$ as a pair $(Z, D)$ satisfying certain properties. We find a modular compactification of such pairs and, in the case of $Z = mathbb{P}^3$ and $D$ a surface, use their properties to classify the pairs on the boundary of the moduli space.
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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.
It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi-Yau threefolds coming from eight planes in $mathbb{P}^3$ does {em not} have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.
In order to obtain existence criteria for orthogonal instanton bundles on $mathbb{P}^n$, we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on $mathbb{P}^n$ and prove that every orthogonal instanton bundle with no global sections on $mathbb{P}^n$ and charge $cgeq 3$ has rank $rleq (n-1)c$. We also prove that when the rank $r$ of the bundles reaches the upper bound, $mathcal{M}_{mathbb{P}^n}^{mathcal{O}}(c,r)$, the coarse moduli space of orthogonal instanton bundles with no global sections on $mathbb{P}^n$, with charge $cgeq 3$ and rank $r$, is affine, reduced and irreducible. Last, we construct Kronecker modules to determine the splitting type of the bundles in $mathcal{M}_{mathbb{P}^n}^{mathcal{O}}(c,r)$, whenever is non-empty.
This paper considers the moduli spaces (stacks) of parabolic bundles (parabolic logarithmic flat bundles with given spectrum, parabolic regular Higgs bundles) with rank 2 and degree 1 over $mathbb{P}^1$ with five marked points. The stratification structures on these moduli spaces (stacks) are investigated. We confirm Simpsons foliation conjecture of moduli space of parabolic logarithmic flat bundles for our case.
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