April 23

Title: Stability conditions for coherent systems on integral curves 

Speaker: Renato Vidal (UFMG)

Abstract: We will briefly introduce the categorical notion of stability, which we apply to the category of coherent systems on integral (irreducible and reduced) projective curves. We define a three-parameter family of pre-stability conditions in its derived category and study when these conditions qualify as true stability conditions. This study is based on some Clifford-type theorems, which can be extended to integral curves. Finally, we discuss some possible applications. This is a joint work with Marcos Jardim and Leonardo Roa-Leguizamón. 

September 30

Title: New infinite series of moduli components of stable rank 3 bundles on rational Fano threefolds of main series

Speaker: Irina Lanskikh, HSE, Moscow

Abstract: We consider the Gieseker–Maruyama moduli spaces of semistable rank 3 coherent sheaves with fixed Chern classes on rational Fano threefolds of the main series, including projective space, the three-dimensional quadric, the intersection of two four-dimensional quadrics, and the linear section of the Grassmannian Gr(2,5). We construct an infinite series of irreducible components of the moduli spaces of rank 3 locally free sheaves. This a joint project with Alexander Tikhomirov.

May 14th

Title: Projectivity of moduli of higher-rank PT-stable pairs on threefolds

Speaker: Mihai-Cosmin Pavel (Simion Stoilow Institute of Mathematics of the Romanian Academy)

Abstract: Stable pairs were introduced by Pandharipande and Thomas in order to define new curve-counting invariants on Calabi–Yau threefolds. It was soon observed (independently by Bayer and Toda) that such objects can be understood via a generalized notion of stability on the derived category of coherent sheaves. This notion, known as Pandharipande–Thomas (PT) stability, extends the original construction and recovers the stable pairs of Pandharipande and Thomas as PT-stable objects of rank 1 and trivial determinant. One is naturally led to study the moduli theory of PT-stable objects on projective threefolds. However, unlike the original case, the moduli problem for higher-rank PT-stable objects is not known to be associated with a GIT problem, and hence it is unknown whether the moduli spaces are projective. In this talk, we present recent progress on this problem, based on joint work with Tuomas Tajakka. 

April 9th

Title: Moduli of rank three sheaves on the projective space P3 with singularities of mixed dimension 

Speaker: Alexander Tikhomirov (HSE, Moscow)

Abstract: We study the Gieseker–Maruyama moduli space of normalized semistable coherent rank three sheaves of positive second Chern class and nonnegative third Chern class on the projective space P3. We find the first example of an irreducible component of this moduli space with small values of Chern classes in which the generic sheaf has singularities of mixed dimension: zero- and one-dimensional singularities simultaneously. Previously, some examples of components of moduli of semistable sheaves with singularities of mixed dimension were constructed only for rank two sheaves by Ivanov and Tikhomirov in 2018 and by Almeida, Jardim, and Tikhomirov in 2022. 

March 19th

Title: The higher rank DT/PT wall-crossing in Bridgeland stability 

Speaker: Cristian Martinez (URosario)

Abstract: It has been almost two decades since Bayer introduced the concept of polynomial stability, in his seminal paper he also showed the existence of two standard polynomial stability conditions separated by a wall called DT and PT for which the rank-1 DT stable objects are precisely the ideal sheaves of curves and the rank-1 PT stable objects are the PT stable pairs (pairs (F,s) consisting of a 1-dimensional sheaf F and a section of F whose cokernel is 0-dimensional).  It has long been suggested that the DT/PT correspondence (a comparison between the virtual counts of DT and PT stable objects) is actually a manifestation of a wall-crossing not only in the realm of polynomial stability conditions but also in the space of Bridgeland stability conditions. For a smooth projective threefold X of Picard number 1, where the generalized Bogomolov-Gieseker inequality conjectured by Bayer, Macrì and Toda holds, we show that for a given Chern character (of any rank) there are two adjacent chambers in the space of geometric stability conditions, where the semistable objects are the Gieseker semistable sheaves and the PT semistable objects, respectively. The contracting map to the wall on one side, turns out to be an analog of the Gieseker-to-Uhlenbeck map for surfaces. This is joint work with Marcos Jardim, Jason Lo, and Antony Maciocia. 

December 3rd

Title: New component of the moduli scheme M_{P3}(2; −1, 4, 0) of semistable rank two sheaves on the projective space

Speaker: Alexander S. Tikhomirov (HSE, Moscow)

Abstract: In this paper we study the Gieseker-Maruyama moduli space MP3 (2; −1, 4, 0) of semistable coherent sheaves of rank two with Chern classes c1 = −1, c2 = 4, c3 = 0 on the projective space P3. To date, only two irreducible components of this space have been known, and their general points are locally free sheaves. In this paper we find and describe a new irreducible component of the space MP3 (2; −1, 4, 0), a general point of which is a sheaf with the singularity at a disjoint union of a pair of lines and a pair of points. We prove that this component has the expected dimension 27 and is generically reduced as a scheme.  (joint project with Mikhail Zavodchikov, Yaroslavl State Univ.) 

November 5th

Title: Stable pairs for rank 2 sheaves on the projective space. 

Speaker: Dapeng Mu (Campinas/Edinburgh)

Links: Youtube

Abstract: A stable pair on a projective variety consists of a sheaf and a global section subject to stability conditions parameterized by a family of rational polynomials. We will show that for a smooth projective threefold and a class of a rank 2 sheaf, there are two stability chambers (in the space of rational polynomials) for which the moduli spaces of semistable pairs are fibered over a Gieseker moduli space of rank 2 semistable sheaves and a Hilbert scheme, respectively. In the latter moduli space, every semistable pair corresponds to a closed subscheme of codimension 2 with an extension class, providing a generalization of the Serre correspondence. These two moduli spaces are related by finitely many wall-crossings. We provide explicit descriptions of those wall-crossings for a certain fixed numerical class. In particular, these wall-crossings preserve the connectedness of the moduli space of semistable pairs. We expect this to be true for any rank 2 class, implying the connectedness for the Gieseker moduli space of rank 2 semistable sheaves provided the corresponding Hilbert scheme is connected.

October 8th

Title: Rank two semistable sheaves with maximal third Chern class on a smooth quadric threefold. 

Speaker: Danil Vassiliev (HSE)

Links: Youtube

Abstract: In this seminar we describe an application of Bridgeland stability and tilt stability to the description of rank two Gieseker semistable sheaves with maximal third Chern class on a smooth quadric threefold. As a byproduct of this description we obtain the first known example of a disconnected moduli space of rank two semistable sheaves with fixed Chern classes on a smooth projective variety.

September 17th