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## Beyond the Sottile-Sturmfels degeneration of a semi-infinite Grassmannian

arxiv.org.
math.
Cornell University
,
2021.
No. 2110.07397 .

We study toric degenerations of semi-infinite Grassmannians (a.k.a. quantum Grassmannians). While the toric degenerations of the classical Grassmannians are well studied, the only known example in the semi-infinite case is due to Sottile-Sturmfels. We start by providing a new interpretation of the Sottile-Sturmfels construction by finding a poset such that their degeneration is the toric variety of the order polytope of the poset. We then use our poset to construct and study a new toric degeneration in the semi-infinite case. Our construction is based on the notion of poset polytopes introduced by Fang-Fourier-Litza-Pegel. As an application we introduce semi-infinite PBW-semistandard tableaux, giving a basis in the homogeneous coordinate ring of a semi-infinite Grassmannian.

Shafarevich A., Results in Mathematics 2021 Vol. 76 No. 3 Article 145

Let KK be an algebraically closed field of characteristic zero and GaGa be the additive group of KK. We say that an irreducible algebraic variety X of dimension n over the field KK admits an additive action if there is a regular action of the group Gna=Ga×⋯×GaGan=Ga×⋯×Ga (n times) on X with an open orbit. In this paper we find all projective toric hypersurfaces admitting additive action. ...

Added: September 10, 2021

Shafarevich A., Moscow University Mathematics Bulletin 2019 Vol. 74 No. 5 P. 209-211

Let X be an affine toric variety over an algebraically closed field of characteristic zero. Orbits of connected component of identity of automorphism group in terms of dimensions of tangent spaces of the variety X are described. A formula to calculate these dimensions is presented. ...

Added: September 10, 2021

Katzarkov L., Mohammed Abouzaid, Auroux D., Publications Mathématiques de l'IHÉS 2016 Vol. 123 No. 1 P. 199-282

https://link.springer.com/article/10.1007/s10240-016-0081-9 ...

Added: October 23, 2017

Galkin S., Donihakalu Shankar Nagaraj, Projective bundles and blow-ups of Projective spaces / Cornell University. Series math "arxiv.org". 2020. No. 2006.12112.

The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces. ...

Added: April 15, 2021

Chebochko N.G., M.I. Kuznetsov, Communications in Algebra 2017 Vol. 45 No. 7 P. 2969-2977

All classes of integrable cocycles in H2(L,L) are obtained for Lie algebra of type G2 over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. ...

Added: October 10, 2017

Alexander I. Efimov, Journal of London Mathematical Society 2014 Vol. 90 No. 2 P. 350-372

In this paper, we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac 34$ there exist infinitely many toric Fano varieties $Y$ with ...

Added: January 28, 2015

Ye L., Faisceau Automorphe Unipotent pour G2,Nombres de Franel, et Stratification deThom-Boardman / Cornell University. Series math "arxiv.org". 2020.

Thesis of the author. ...

Added: December 16, 2019

Feigin E., Makhlin I., Semitoric degenerations of Hibi varieties and flag varieties / Cornell University. Series math "arxiv.org". 2020. No. 2008.13243.

We construct a family of flat semitoric degenerations for the Hibi variety of every finite distributive lattice. The irreducible components of each degeneration are the toric varieties associated with polytopes forming a regular subdivision of the order polytope of the underlying poset. These components are themselves Hibi varieties. For each degeneration in our family we ...

Added: September 1, 2020

Kuznetsov A., Alexander Polishchuk, Journal of the European Mathematical Society 2016 Vol. 18 No. 3 P. 507-574

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups. ...

Added: December 22, 2013

Galkin S., Mellit A., Smirnov M., Dubrovin's conjecture for IG(2,6) / Cornell University. Series math "arxiv.org". 2014. No. 1405.3857.

We show that the big quantum cohomology of the symplectic isotropic Grassmanian IG(2,6) is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture holds and stresses the need to consider the big quantum cohomology in its formulation. ...

Added: May 16, 2014

Dumanski I., Feigin E., Finkelberg M. V., Beilinson-Drinfeld Schubert varieties and global Demazure modules / Cornell University. Series math "arxiv.org". 2020. No. 2003.12930.

We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson- Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules of the current Lie algebra. ...

Added: April 2, 2020

Bodzenta-Skibinska A., DG quivers of smooth rational surfaces / Cornell University. Series math "arxiv.org". 2013.

Let X be a smooth rational surface. We calculate a DG quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate canonical DG algebras of smooth toric surfaces. ...

Added: November 5, 2014

Ivan Penkov, Tikhomirov A. S., Pure and Applied Mathematics Quarterly 2014 Vol. 10 No. 2 P. 289-323

We consider ind-varieties obtained as direct limits of chains of embeddings $X_1\stackrel{\phi_1}{\hookrightarrow}\dots\stackrel{\phi_{m-1}}{\hookrightarrow} X_m\stackrel{\phi_m}{\hookrightarrow}X_{m+1}\stackrel{\phi_{m+1}}{\hookrightarrow}\dots$, where each $X_m$ is a grassmannian or an isotropic grassmannian (possibly mixing grassmannians and isotropic grassmannians), and the embeddings $\phi_m$ are linear in the sense that they induce isomorphisms of Picard groups. We prove that any such ind-variety is isomorphic to one ...

Added: October 9, 2014

Илья Александрович Болдырев, Gayfullin S., Математические заметки 2021 Т. 110 № 6 С. 837-855

Criteria for the flexibility, rigidity, and almost rigidity of nonnormal affine toric varieties are obtained. For rigid and almost rigid toric varieties, automorphism groups are explicitly calculated. ...

Added: February 6, 2022

Kuznetsov A., Alexander Polishchuk, Exceptional collections on isotropic Grassmannians / Cornell University. Series math "arxiv.org". 2011. No. 1110.5607 .

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups. ...

Added: October 4, 2013

Fonarev A., Moscow Mathematical Journal 2016 Vol. 16 No. 4 P. 711-726

We classify irreducible equivariant Ulrich vector bundles on isotropic Grassmannians. ...

Added: November 7, 2017

Galkin S., Pieter Belmans, Swarnava Mukhopadhyay, Experimental Mathematics 2019

We give the first examples of smooth Fano and Calabi–Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarized varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have ...

Added: October 4, 2019

Stanislav Fedotov, Transactions of the American Mathematical Society 2013 Vol. 365 No. 8 P. 4153-4179

In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Marcus Reineke. Obtained is a generalization of this construction for finite dimensional associative algebras and for quivers with oriented cycles. As an application we get an explicit realization of fibers for the moduli space bundle ...

Added: November 5, 2015

Anton Fonarev, On the bounded derived category of $\mathsf{IGr}(3, 7)$ / Cornell University. Series arXiv "math". 2018.

We construct a mininal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian $\IGr(3,7)$. Moreover, we show that $\IGr(3, 7)$ admits a full exceptional collection consisting of equivariant vector bundles. ...

Added: April 20, 2018

Galkin S., Mellit A., Smirnov M., International Mathematics Research Notices 2015 Vol. 2015 No. 18 P. 8847-8859

We show that the big quantum cohomology of the symplectic isotropic Grassmanian IG(2,6) is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture holds and stresses the need to consider the big quantum cohomology in its formulation. ...

Added: October 20, 2014

Przyjalkowski V., Shramov K., Bulletin of the Korean Mathematical Society 2017 Vol. 54 No. 5 P. 1527-1575

In a spirit of Givental's constructions Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested Landau--Ginzburg models for smooth Fano complete intersections in Grassmannians and partial flag varieties as certain complete intersections in complex tori equipped with special functions called superpotentials. We provide a particular algorithm for constructing birational isomorphisms of these models for complete intersections in ...

Added: October 8, 2019

Lvovsky S., М.: МЦНМО, 2013

Всякое одномерное семейство прямых на плоскости (кроме вырожденных случаев) является семейством касательных к некоторой кривой. В пространстве, однако, это уже совершенно не так; в брошюре объясняется, как, глядя на одномерное семейство прямых в пространстве, определить, является ли оно «касательным». По ходу дела чита- тель знакомится с такими важными понятиями современной математики, как внешняя алгебра и ...

Added: October 3, 2013

Przyjalkowski V., Shramov K., Успехи математических наук 2014 Т. 69 № 6(420) С. 181-182

В Московском математическом обществе.
Сообщения Московского математического общества ...

Added: February 26, 2015

Kuznetsov A., Smirnov M., Proceedings of the London Mathematical Society 2020 Vol. 120 No. 5 P. 617-641

We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the Grassmannian G(k,n), we conjecture that the residual category associated with Fonarev’s Lefschetz exceptional collection is generated by a completely orthogonal exceptional collection. We prove this conjecture for k = p, ...

Added: August 12, 2020