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On the irreducibility of Hilbert scheme of surfaces of minimal degree

100%
Bogomolov F., Kulikov V.
Open Mathematics
|
2013
|
tom 11
|
nr 2
254-263
EN
The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.
2
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Quotients of an affine variety by an action of a torus

61%
Chuvashova O., Pechenkin N.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1863-1880
EN
Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/C T and the toric Hilbert scheme H. We introduce a notion of the main component H 0 of H, which parameterizes general T-orbit closures in X and their flat limits. The main component U 0 of the universal family U over H is a preimage of H 0. We define an analogue of a universal family WX over the main component of X/C T. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U 0 to W X. The variety W X also provides a geometric realization of the Altmann-Hausen family. In particular, the notion of W X allows us to provide an explicit description of the fan of the Altmann-Hausen family in the toric case.
3
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Elliptic curves on spinor varieties

61%
Perrin N.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1393-1406
EN
We prove irreducibility of the scheme of morphisms, of degree large enough, from a smooth elliptic curve to spinor varieties. We give an explicit bound on the degree.
4
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Transverse Hilbert schemes and completely integrable systems

52%
Donin N. L. L.
Complex Manifolds
|
2017
|
tom 4
|
nr 1
263-272
EN
In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].
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