Stein Arild Strømme’s research while affiliated with University of Bergen and other places

... The issue regarding the rationality of the moduli spaces of instanton bundles was raised by Hartshorne [25], and turned out to be a difficult problem. So far, it's known that these varieties are rational for rank-2 and charge n " 2, 4, 5, due to works of Hirschowitz-Narasimhan [28], Ellingsrud-Strømme [21], and Katsylo [35]; for ně6, the issue remained open, in spite of recent efforts [41,42]. Beorchia-Franco [8] proved that the moduli space of 't Hooft instantons -those which possess a section at the first twist-is irreducible and rational. ...

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Rationality and categorical properties of the moduli of instanton bundles on the projective 3-space

... Roughly, it states that the Gromov-Witten theory of the 'nonabelian space' V //G is related to that of the 'abelian space' V //T, where T is a maximal torus in G. The classical version goes back to Ellingsrud and Strømme [1], and Martin [2], and the quantum formulation is due to Bertram, Ciocan-Fontanine, and Kim [3,4] and Ciocan-Fontanine, Kim, and Sabbah [5]. In this paper we consider the GIT quotient X to be the Grassmannian Gr(k; N). ...

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A Correspondence between the Quantum K Theory and Quantum Cohomology of Grassmannians

... By choosing a = (−1, 1) (see [BBF + 23] for the notation), we have a universal representation consisting of two universal bundles and a universal representation map V 1 → V 2 ⊗ W . It turns out that V 1 = U 1 and V 2 = U 2 (see [ES95]). By definition, we have the natural inclusion of vector bundles over Y : U 1 ֒→ S 2,1 W and U 2 ֒→ S 2 W . ...

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Quantum Cohomology of a Fano Quiver Moduli Space

... Proof. By construction (using GIT for example), M := M (r, c 1 , c 2 ) is a smooth projective variety defined over R. Since M is a fine moduli space, there is a universal sheaf E on M × P 2 , which is also defined over R. By Ellingsrud and Strømme [ES93], the cohomology ring H * (M, Z) has no odd cohomology, and it is torsion-free and generated by the elements ...

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Maximal real varieties from moduli constructions

... We conclude this section by recalling the following theorem by Beauville [Bea95]; see also Ellingsrud-Strømme [ES93]. ...

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Generators for the cohomology ring of the moduli of 1-dimensional sheaves on $\mathbb{P}^2

... Let Quot(E, P ) denote the Quot scheme parametrizing quotients of E with Hilbert polynomial P . In [Str87], when C = P 1 , the quot scheme Quot(Q ⊕n C , P ) is studied. In particular, its nef cone is described. ...

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Some recent results on the punctual Quot schemes

... Hilbert schemes of surfaces we refer to [33,19,59], whereas for combinatorial aspects on 3-folds (in the language of plane partitions) we refer to [2,1] and finally we refer to the recent work [38] for a combinatorial advance in all dimensions. ...

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Invariants of nested Hilbert and Quot schemes on surfaces

... It is well known that the moduli space of semistable bundles with fixed invariants exists, and that sometimes is fine. Some moduli spaces of unstable bundles have been constructed by adding some extra information: the moduli spaces of unstable vector bundles of rank 2 and 3 were constructed in [14] and in [38], respectively, when dim X = 1 and the algebra of endomorphisms is fixed; in [40] when X is the projective plane and in [3] when X = P 3 (C), in both cases they consider the degree of instability. J. M. Drezet studied in [19] the case of very unstable bundles when dim X > 2. In [26] the authors construct the moduli spaces of pure sheaves with fixed Harder-Narasimhan type which have some additional data called an m-rigidification. ...

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Moduli of unstable bundles of HN-length two with fixed algebra of endomorphisms

... However the most well-studied examples are vector bundles on P n , especially rank 2 vector bundles on P 2 . There is the classical theory of jumping lines (which has too many applications to list here) and also work on the jumping locus of higher degree curves ( [Sm84], [Man90], [Ran01], [Vit04], [Mar24]). Much of the previous work has been example-based, studying the jumping loci for particular vector bundles and observing whether they did or did not have the expected dimension. ...

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Codimension of jumping loci

... The main subject of this paper is the Calogero-Moser spaces  . In Section 3, we discuss some of their properties and their topology; it turns out [33] that  is diffeomorphic to the Hilbert scheme of points in the affine plane ℂ 2 whose topology has been fully described by Ellingsrud and Strømme [12,13]. Since its first de Rham cohomology group vanishes, every holomorphic symplectic vector field on  is Hamiltonian. ...

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The symplectic density property for Calogero–Moser spaces