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Journal articles on the topic 'Grassmannian varieties'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

MARTÍN, FRANCISCO J. PLAZA. "PRYM VARIETIES AND THE INFINITE GRASSMANNIAN." International Journal of Mathematics 09, no. 01 (February 1998): 75–93. http://dx.doi.org/10.1142/s0129167x98000051.

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In this paper we study Prym varieties and their moduli space using the well-known techniques of the infinite Grassmannian. There are three main results of this paper: a new definition of the BKP hierarchy over an arbitrary base field (that generalizes the classical one over [Formula: see text]); a characterization of Prym varieties in terms of dynamical systems, and explicit equations for the moduli space of (certain) Prym varieties. For all of these problems the language of the infinte Grassmannian, in its algebro-geometric version, allows us to deal with these problems from the same point of view.

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2

Draisma, Jan, and Rob H. Eggermont. "Plücker varieties and higher secants of Sato’s Grassmannian." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 737 (April 1, 2018): 189–215. http://dx.doi.org/10.1515/crelle-2015-0035.

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AbstractEvery Grassmannian, in its Plücker embedding, is defined by quadratic polynomials. We prove a vast, qualitative, generalisation of this fact to what we callPlücker varieties. A Plücker variety is in fact a family of varieties in exterior powers of vector spaces that, like the Grassmannian, is functorial in the vector space and behaves well under duals. A special case of our result says that for each fixed natural numberk, thek-th secant variety ofanyPlücker-embedded Grassmannian is defined in bounded degree independent of the Grassmannian. Our approach is to take the limit of a Plücker variety in the dual of a highly symmetric space known as theinfinite wedge, and to prove that up to symmetry the limit is defined by finitely many polynomial equations. For this we prove the auxiliary result that for every natural numberpthe space ofp-tuples of infinite-by-infinite matrices is Noetherian modulo row and column operations. Our results have algorithmic counterparts: every bounded Plücker variety has a polynomial-time membership test, and the same holds for Zariski-closed, basis-independent properties ofp-tuples of matrices.

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3

Shimada, Ichiro. "Zariski Hyperplane Section Theorem for Grassmannian Varieties." Canadian Journal of Mathematics 55, no. 1 (February 1, 2003): 157–80. http://dx.doi.org/10.4153/cjm-2003-007-9.

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AbstractLet ϕ: X → M be a morphism from a smooth irreducible complex quasi-projective variety X to a Grassmannian variety M such that the image is of dimension ≥ 2. Let D be a reduced hypersurface in M, and γ a general linear automorphism of M. We show that, under a certain differentialgeometric condition on ϕ(X) and D, the fundamental group π1((γ ○ ϕ)−1 (M \ D)) is isomorphic to a central extension of π1(M \ D) × π1(X) by the cokernel of π2(ϕ) : π2(X) → π2(M).

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4

Braun, Lukas. "Hilbert series of the Grassmannian and k-Narayana numbers." Communications in Mathematics 27, no. 1 (June 1, 2019): 27–41. http://dx.doi.org/10.2478/cm-2019-0003.

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AbstractWe compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the q-Hilbert series is a Vandermonde-like determinant. We show that the h-polynomial of the Grassmannian coincides with the k-Narayana polynomial. A simplified formula for the h-polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the k-Narayana numbers, i.e. the h-polynomial of the Grassmannian.

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5

LENAGAN, T. H., and L. RIGAL. "QUANTUM ANALOGUES OF SCHUBERT VARIETIES IN THE GRASSMANNIAN." Glasgow Mathematical Journal 50, no. 1 (January 2008): 55–70. http://dx.doi.org/10.1017/s0017089507003928.

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AbstractWe study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.

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6

Serhiyenko, K., M. Sherman‐Bennett, and L. Williams. "Cluster structures in Schubert varieties in the Grassmannian." Proceedings of the London Mathematical Society 119, no. 6 (July 29, 2019): 1694–744. http://dx.doi.org/10.1112/plms.12281.

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7

Bakshi, Sarjick, S. Senthamarai Kannan, and Subrahmanyam K. Venkata. "Torus quotients of Richardson varieties in the Grassmannian." Communications in Algebra 48, no. 2 (September 27, 2019): 891–914. http://dx.doi.org/10.1080/00927872.2019.1668005.

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8

SCHILLEWAERT, J., and H. VAN MALDEGHEM. "A COMBINATORIAL CHARACTERIZATION OF THE LAGRANGIAN GRASSMANNIAN LG(3,6)()." Glasgow Mathematical Journal 58, no. 2 (July 21, 2015): 293–311. http://dx.doi.org/10.1017/s0017089515000208.

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AbstractWe provide a combinatorial characterization of LG(3,6)(${\mathbb{K}}$) using an axiom set which is the natural continuation of the Mazzocca–Melone set we used to characterize Severi varieties over arbitrary fields (Schillewaert and Van Maldeghem, Severi varieties over arbitrary fields,Preprint). This fits within a large project aiming at constructing and characterizing the varieties related to the Freudenthal–Tits magic square.

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9

Boe, Brian D., and Joseph H. G. Fu. "Characteristic Cycles in Hermitian Symmetric Spaces." Canadian Journal of Mathematics 49, no. 3 (June 1, 1997): 417–67. http://dx.doi.org/10.4153/cjm-1997-021-7.

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AbstractWe give explicit combinatorial expresssions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties X in the classical Hermitian symmetric spaces: namely the intersection homology sheaves IHX and the constant sheaves ℂX. The three main cases of interest are the Hermitian symmetric spaces for groups of type An (the standard Grassmannian), Cn (the Lagrangian Grassmannian) and Dn. In particular we find that CC(IHX) is irreducible for all Schubert varieties X if and only if the associated Dynkin diagramis simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the Cn and Dn cases are new.Our approach is to compute CC(ℂX) by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute CC(IHX). The geometric method is based on the fundamental formula where the Xr ↓ X constitute a family of tubes around the variety X. This formula leads at once to an expression for the coefficients of CC(ℂX) as the degrees of certain singular maps between spheres.

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10

Upadhyay, Shyamashree. "Initial Ideals of Tangent Cones to the Richardson Varieties in the Orthogonal Grassmannian." International Journal of Combinatorics 2013 (March 26, 2013): 1–19. http://dx.doi.org/10.1155/2013/392437.

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A Richardson variety in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian and an opposite Schubert variety therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T-fixed point of , thus generalizing a result of Raghavan and Upadhyay (2009). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal-bounded-RSK (OBRSK).

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11

SHIMADA, ICHIRO. "GENERALIZED ZARISKI–VAN KAMPEN THEOREM AND ITS APPLICATION TO GRASSMANNIAN DUAL VARIETIES." International Journal of Mathematics 21, no. 05 (May 2010): 591–637. http://dx.doi.org/10.1142/s0129167x10006252.

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We formulate and prove a generalization of Zariski–van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz–Zariski–van Kampen type for the fundamental groups of the complements to the Grassmannian dual varieties.

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12

González-Alonso, Víctor. "Grassmannian BGG complexes and Hodge numbers of irregular varieties." Bulletin of the London Mathematical Society 47, no. 5 (August 6, 2015): 743–58. http://dx.doi.org/10.1112/blms/bdv051.

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13

Ballico, E. "Joins and secant varieties of subvarieties of a Grassmannian." Results in Mathematics 32, no. 1-2 (August 1997): 29–36. http://dx.doi.org/10.1007/bf03322521.

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14

G�mez Gonz�lez, E., J. M. Mu�oz Porras, and F. J. Plaza Mart�n. "Prym varieties, curves with automorphisms and the Sato Grassmannian." Mathematische Annalen 327, no. 4 (December 1, 2003): 609–39. http://dx.doi.org/10.1007/s00208-003-0422-0.

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15

Crooks, Peter, and Tyler Holden. "Generalized Equivariant Cohomology and Stratifications." Canadian Mathematical Bulletin 59, no. 3 (September 1, 2016): 483–96. http://dx.doi.org/10.4153/cmb-2016-032-5.

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AbstractFor T a compact torus and a generalized T-equivariant cohomology theory, we provide a systematic framework for computing in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute as an (pt)-module when X is a direct limit of smooth complex projective Tℂ-varieties. We perform this computation on the affine Grassmannian of a complex semisimple group.

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16

HWANG, JUN-MUK. "VARIETIES WITH DEGENERATE GAUSS MAPPINGS IN COMPLEX HYPERBOLIC SPACE FORMS." International Journal of Mathematics 13, no. 02 (March 2002): 209–16. http://dx.doi.org/10.1142/s0129167x02001186.

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In analogy with the Gauss mapping for a subvariety in the complex projective space, the Gauss mapping for a subvariety in a complex hyperbolic space form can be defined as a map from the smooth locus of the subvariety to the quotient of a suitable domain in the Grassmannian. For complex hyperbolic space forms of finite volume, it is proved that the Gauss mapping is degenerate if and only if the subvariety is totally geodesic.

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17

Ghorpade, Sudhir R., and K. N. Raghavan. "Hilbert functions of points on Schubert varieties in the symplectic Grassmannian." Transactions of the American Mathematical Society 358, no. 12 (December 1, 2006): 5401–24. http://dx.doi.org/10.1090/s0002-9947-06-04037-2.

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18

MUKHIN, E., and A. VARCHENKO. "CRITICAL POINTS OF MASTER FUNCTIONS AND FLAG VARIETIES." Communications in Contemporary Mathematics 06, no. 01 (February 2004): 111–63. http://dx.doi.org/10.1142/s0219199704001288.

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We consider critical points of master functions associated with integral dominant weights of Kac–Moody algebras and introduce a generating procedure constructing new critical points starting from a given one. The set of all critical points constructed from a given one is called a population. We formulate a conjecture that a population is isomorphic to the flag variety of the Langlands dual Kac–Moody algebra and prove the conjecture for algebras slN+1, so2N+1, and sp2N. We show that populations associated with a collection of integral dominant slN+1-weights are in one to one correspondence with intersection points of suitable Schubert cycles in a Grassmannian variety.

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19

Bernardi, Alessandra, Enrico Carlini, Maria Catalisano, Alessandro Gimigliano, and Alessandro Oneto. "The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition." Mathematics 6, no. 12 (December 8, 2018): 314. http://dx.doi.org/10.3390/math6120314.

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We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

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20

Rigal, L., and P. Zadunaisky. "Quantum analogues of Richardson varieties in the grassmannian and their toric degeneration." Journal of Algebra 372 (December 2012): 293–317. http://dx.doi.org/10.1016/j.jalgebra.2012.09.016.

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21

Knutson, Allen, Thomas Lam, and David E. Speyer. "Positroid varieties: juggling and geometry." Compositio Mathematica 149, no. 10 (August 19, 2013): 1710–52. http://dx.doi.org/10.1112/s0010437x13007240.

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AbstractWhile the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.

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22

Jiang, Yuhan, and Bernd Sturmfels. "Bad projections of the PSD cone." Collectanea Mathematica 72, no. 2 (March 29, 2021): 261–80. http://dx.doi.org/10.1007/s13348-021-00319-4.

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AbstractThe image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.

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23

Catanese, F. "Cayley Forms and Self-Dual Varieties." Proceedings of the Edinburgh Mathematical Society 57, no. 1 (December 19, 2013): 89–109. http://dx.doi.org/10.1017/s0013091513000928.

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AbstractGeneralized Chow forms were introduced by Cayley for the case of 3-space; their zero set on the Grassmannian G(1, 3) is either the set Z of lines touching a given space curve (the case of an ‘honest’ Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equation Q for G(1, 3). Our main result is that F is a Cayley form if and only if Z = G(1, 3) ∩ {F = 0} is equal to its dual variety. We also show that the variety of generalized Cayley forms is defined by quadratic equations, since there is a unique representative F0 + QF1 of F, with F0, F1 harmonic, such that the harmonic projection of the Cayley equation is identically 0. We also give new equations for honest Cayley forms, but show, with some calculations, that the variety of honest Cayley forms does not seem to be defined by quadratic and cubic equations.

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24

Crooks, Peter, and Steven Rayan. "Some results on equivariant contact geometry for partial flag varieties." International Journal of Mathematics 27, no. 08 (July 2016): 1650066. http://dx.doi.org/10.1142/s0129167x1650066x.

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We study equivariant contact structures on complex projective varieties arising as partial flag varieties [Formula: see text], where [Formula: see text] is a connected, simply-connected complex simple group of type ADE and [Formula: see text] is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby’s classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of [Formula: see text]-equivariant vector bundles on [Formula: see text]. A byproduct of our argument is a canonical, global description of the unique [Formula: see text]-invariant contact structure on the isotropic Grassmannian of 2-planes in [Formula: see text].

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25

Choe, Insong, and George H. Hitching. "Non-defectivity of Grassmannian bundles over a curve." International Journal of Mathematics 27, no. 07 (June 2016): 1640002. http://dx.doi.org/10.1142/s0129167x16400024.

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Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

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26

Thomas, Hugh, and Alexander Yong. "Multiplicity-Free Schubert Calculus." Canadian Mathematical Bulletin 53, no. 1 (March 1, 2010): 171–86. http://dx.doi.org/10.4153/cmb-2010-032-x.

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AbstractMultiplicity-free algebraic geometry is the study of subvarieties Y ⊆ X with the “smallest invariants” as witnessed by a multiplicity-free Chow ring decomposition of [Y] ∈ A*(X) into a predetermined linear basis.This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.

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27

Hiller, Howard. "On the cohomology of loop spaces of compact Lie groups." Glasgow Mathematical Journal 26, no. 1 (January 1985): 91–99. http://dx.doi.org/10.1017/s0017089500005814.

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Let G be a compact, simply-connected Lie group. The cohomology of the loop space ΏG has been described by Bott, both in terms of a cell decomposition [1] and certain homogeneous spaces called generating varieties [2]. It is possible to view ΏG as an infinite dimensional “Grassmannian” associated to an appropriate infinite dimensional group, cf. [3], [7]. From this point of view the above cell-decomposition of Bott arises from a Bruhat decomposition of the associated group. We choose a generator H ∈ H2(ΏG, ℤ) and call it the hyperplane class. For a finite-dimensional Grassmannian the highest power of H carries geometric information about the variety, namely, its degree. An analogous question for ΏG is: What is the largest integer Nk = Nk(G) which divides Hk ∈ H2k(ΏG, ℤ)?Of course, if G = SU(2) = S3, one knows Nk = h!. In general, the deviation of Nk from k! measures the failure of H to generate a divided polynomial algebra in H*(ΏG, ℤ).

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28

KAMNITZER, JOEL, DINAKAR MUTHIAH, and ALEX WEEKES. "ON A REDUCEDNESS CONJECTURE FOR SPHERICAL SCHUBERT VARIETIES AND SLICES IN THE AFFINE GRASSMANNIAN." Transformation Groups 23, no. 3 (November 13, 2017): 707–22. http://dx.doi.org/10.1007/s00031-017-9455-4.

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29

Kreiman, Victor. "Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence." Journal of Algebraic Combinatorics 27, no. 3 (September 18, 2007): 351–82. http://dx.doi.org/10.1007/s10801-007-0093-0.

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30

Konopelchenko, B. G., and G. Ortenzi. "Algebraic varieties in the Birkhoff strata of the Grassmannian Gr(2): Harrison cohomology and integrable systems." Journal of Physics A: Mathematical and Theoretical 44, no. 46 (October 21, 2011): 465201. http://dx.doi.org/10.1088/1751-8113/44/46/465201.

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31

Littig, Peter J., and Stephen A. Mitchell. "Generating varieties for affine Grassmannians." Transactions of the American Mathematical Society 363, no. 07 (July 1, 2011): 3717. http://dx.doi.org/10.1090/s0002-9947-2011-05257-8.

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32

Cerulli Irelli, Giovanni, Evgeny Feigin, and Markus Reineke. "Quiver Grassmannians and degenerate flag varieties." Algebra & Number Theory 6, no. 1 (June 15, 2012): 165–94. http://dx.doi.org/10.2140/ant.2012.6.165.

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33

Hefez, A., and A. Thorup. "Reflexivity of grassmannians and segre varieties." Communications in Algebra 15, no. 6 (January 1987): 1095–108. http://dx.doi.org/10.1080/00927878708823458.

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34

Ballico, E., and A. Cossidente. "Cones over Veronese varieties and Grassmannians." Results in Mathematics 38, no. 3-4 (November 2000): 209–12. http://dx.doi.org/10.1007/bf03322008.

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35

Kinser, Ryan, and Jenna Rajchgot. "Type D quiver representation varieties, double Grassmannians, and symmetric varieties." Advances in Mathematics 376 (January 2021): 107454. http://dx.doi.org/10.1016/j.aim.2020.107454.

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36

Lima-Filho, Paulo, and Hal Schenck. "Efficient computation of resonance varieties via Grassmannians." Journal of Pure and Applied Algebra 213, no. 8 (August 2009): 1606–11. http://dx.doi.org/10.1016/j.jpaa.2008.11.021.

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37

Smirnov, E. Yu. "Desingularizations of Schubert varieties in double Grassmannians." Functional Analysis and Its Applications 42, no. 2 (April 2008): 126–34. http://dx.doi.org/10.1007/s10688-008-0018-2.

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38

Nanduri, Ramakrishna. "Hilbert coefficients of Schubert varieties in Grassmannians." Journal of Algebra and Its Applications 14, no. 03 (November 7, 2014): 1550036. http://dx.doi.org/10.1142/s021949881550036x.

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In this paper, we give combinatorial formulas for the Hilbert coefficients, h-polynomial and the Cohen–Macaulay type of Schubert varieties in Grassmannians in terms of the posets associated with them. As a consequence, necessary conditions for a Schubert variety to be a complete intersection and combinatorial criteria are given for a Schubert variety to be Gorenstein and almost Gorenstein, respectively.

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39

Billey, Sara C., and Stephen A. Mitchell. "Smooth and palindromic Schubert varieties in affine Grassmannians." Journal of Algebraic Combinatorics 31, no. 2 (May 23, 2009): 169–216. http://dx.doi.org/10.1007/s10801-009-0181-4.

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40

Haines, Thomas J., and Timo Richarz. "Smoothness of Schubert varieties in twisted affine Grassmannians." Duke Mathematical Journal 169, no. 17 (November 2020): 3223–60. http://dx.doi.org/10.1215/00127094-2020-0025.

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41

Berget, Andrew, and Jia Huang. "Cyclic sieving of finite Grassmannians and flag varieties." Discrete Mathematics 312, no. 5 (March 2012): 898–910. http://dx.doi.org/10.1016/j.disc.2011.10.015.

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42

Ringel, Claus Michael. "Quiver Grassmannians and Auslander varieties for wild algebras." Journal of Algebra 402 (March 2014): 351–57. http://dx.doi.org/10.1016/j.jalgebra.2013.12.021.

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43

Keller, Bernhard, and Sarah Scherotzke. "Desingularizations of quiver Grassmannians via graded quiver varieties." Advances in Mathematics 256 (May 2014): 318–47. http://dx.doi.org/10.1016/j.aim.2014.01.021.

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44

Savage, Alistair, and Peter Tingley. "Quiver grassmannians, quiver varieties and the preprojective algebra." Pacific Journal of Mathematics 251, no. 2 (June 3, 2011): 393–429. http://dx.doi.org/10.2140/pjm.2011.251.393.

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45

Borisov, Lev A., Andrei Căldăraru, and Alexander Perry. "Intersections of two Grassmannians in ℙ9." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 760 (March 1, 2020): 133–62. http://dx.doi.org/10.1515/crelle-2018-0014.

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AbstractWe study the intersection of two copies of {\mathrm{Gr}(2,5)} embedded in {{{\mathbb{P}}}^{9}}, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi–Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi–Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of non-birationality involves a detailed study of the moduli stack of Calabi–Yau threefolds of the above type, which may be of independent interest.

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46

Gille, Philippe, and Nikita Semenov. "Zero-cycles on projective varieties and the norm principle." Compositio Mathematica 146, no. 2 (December 23, 2009): 457–64. http://dx.doi.org/10.1112/s0010437x09004394.

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AbstractUsing the Gille–Merkurjev norm principle we compute in a uniform way the image of the degree map for quadrics (Springer’s theorem), for twisted forms of maximal orthogonal Grassmannians (a theorem of Bayer-Fluckiger and Lenstra), and for E6- (a theorem of Rost) and E7-varieties.

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47

Mirković, Ivan, and Maxim Vybornov. "On quiver varieties and affine Grassmannians of type A." Comptes Rendus Mathematique 336, no. 3 (February 2003): 207–12. http://dx.doi.org/10.1016/s1631-073x(03)00022-0.

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48

Kodiyalam, Vijay, and K. N. Raghavan. "Hilbert functions of points on Schubert varieties in Grassmannians." Journal of Algebra 270, no. 1 (December 2003): 28–54. http://dx.doi.org/10.1016/s0021-8693(03)00518-0.

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49

HONG, JAEHYUN. "CLASSIFICATION OF SMOOTH SCHUBERT VARIETIES IN THE SYMPLECTIC GRASSMANNIANS." Journal of the Korean Mathematical Society 52, no. 5 (September 1, 2015): 1109–22. http://dx.doi.org/10.4134/jkms.2015.52.5.1109.

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50

Bossinger, Lara, Bosco Frías-Medina, Timothy Magee, and Alfredo Nájera Chávez. "Toric degenerations of cluster varieties and cluster duality." Compositio Mathematica 156, no. 10 (October 2020): 2149–206. http://dx.doi.org/10.1112/s0010437x2000740x.

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We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

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