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Fourier-Mukai partners for very general special cubic fourfolds
No full textWe exhibit explicit examples of very general special cubic fourfolds with discriminant d admitting an associated (twisted) K3 surface, which have non-isomorphic Fourier-Mukai partners. In particular, in the untwisted setting, we show that the number of Fourier-Mukai partners for a very general special cubic fourfold with discriminant d and having an associated K3 surface, is equal to the number m of Fourier-Mukai partners of its associated K3 surface, if d equivalent to 2(mod 6); else, if d equivalent to 0(mod 6), the cubic fourfold has inverted right perpendicularm/2inverted left perpendicular Fourier-Mukai partners
On the double EPW sextic associated to a Gushel-Mukai fourfold
Get PDFIn analogy to the case of cubic fourfolds, we discuss the conditions under which the double cover ilde{Y}_A of the EPW sextic hypersurface associated to a Gushel-Mukai fourfold is birationally equivalent to a moduli space of (twisted) stable sheaves on a K3 surface. In particular, we prove that ilde{Y}_A is birational to the Hilbert scheme of two points on a K3 surface if and only if the Gushel-Mukai fourfold is Hodge-special with discriminant d such that the negative Pell equation P_{d/2}(-1) is solvable in mathbb{Z}
Some remarks on Fano threefolds of index two and stability conditions
Get PDFWe prove that ideal sheaves of lines in a Fano three-fold X of Picard rank one and index two are stable objects in the Kuznetsov component Ku(X), with respect to the stability conditions constructed by Bayer, Lahoz, Macrì, and Stellari, giving a modular description to the Hilbert scheme of lines in X. When X is a cubic three-fold, we show that the Serre functor of Ku(X) preserves these stability conditions. As an application, we obtain the smoothness of nonempty moduli spaces of stable objects in Ku(X). When X is a quartic double solid, we describe a connected component of the stability manifold parametrizing stability conditions on Ku(X)
Voevodsky's conjecture for cubic fourfolds and Gushel-Mukai fourfolds via noncommutative K3 surfaces
Get PDFIn the first part of this paper we will prove the Voevodsky’s nilpotence conjecture for smooth cubic fourfolds and ordinary generic Gushel-Mukai fourfolds. Then, making use of noncommutative motives, we will prove the Voevodsky’s nilpotence conjecture for generic Gushel-Mukai fourfolds containing a au-plane Gr(2, 3) and for ordinary Gushel-Mukai fourfolds containing a quintic del Pezzo surface
Some remarks about deformation theory and formality conjecture
Get PDFUsing the algebraic criterion proved by Bandiera, Manetti and Meazzini, we show the formality conjecture for universally gluable objects with linearly reductive automorphism groups in the bounded derived category of a K3 surface. As an application, we prove the formality conjecture for polystable objects in the Kuznetsov components of Gushel–Mukai threefolds and quartic double solids
Marked and labelled Gushel-Mukai fourfolds
No full textWe prove that the moduli stacks of marked and labelled Hodge-special Gushel–Mukai fourfolds are isomorphic. As an application, we construct rational maps from the stack of Hodge-special Gushel–Mukai fourfolds of discriminant d to the moduli space of (twisted) degree-d polarized K3 surfaces. We use these results to prove a counting formula for the number of 4-dimensional fibers of Fourier–Mukai partners of very general Hodge-special Gushel–Mukai fourfolds with associated K3 surface, and a lower bound for this number in the case of a twisted associated K3 surface
Categorical Torelli theorems: results and open problems
Get PDFWe survey some recent results concerning the so called Categorical Torelli problem. This
is to say how one can reconstruct a smooth projective variety up to isomorphism, by using
the homological properties of special admissible subcategories of the bounded derived cat-
egory of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano
threefolds and cubic fourfolds
Twisted cubics on cubic fourfolds and stability conditions
Get PDFWe give an interpretation of the Fano variety of lines on a cubic fourfold and of the hyperk¨ahler eightfold, constructed by Lehn, Lehn, Sorger and van Straten from twisted cubic curves in a cubic fourfold not containing a plane, as moduli spaces of Bridgeland stable objects in the Kuznetsov component. As a consequence, we obtain the identification of the period point of the LLSvS eightfold with that of the Fano variety. We discuss the derived Torelli theorem for cubic fourfolds
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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