arXiv subjects
Publications and source records attributed to Minyoung Jeon.
In this note, we show that the Hilbert scheme $\text{Hilb}_{P_{d,4}(t)}(E_6/P_6)$ associated with the Hilbert polynomial $P_{d,4}(t)$ is disconnected by determining that it has exactly two connected components. This result adapts Seong's methods, successfully extending the disconnectedness of Grassmannians to the exceptional type $E_6$.
We show the number of connected components of the Hilbert scheme of orthogonal Grassmannians under certain condition, and use this result to describe the geometry of the Hilbert scheme. Subsequently, we determine the Nef cone of the Hilbert scheme by identifying curves dual to the generators of its Neron-Severi group. Our results generalize those of ordinary Grassmannians by Seong, and our approach adapts his proof technique alongside relevant Schubert calculus.
Let $G = GL(n)$ and $K = GL(p) \times GL(q)$ with $p+q=n$, where the groups are taken over $\C$. In this paper we study a certain family of $K$-orbit closures on the flag variety $X$ of $G$. The geometry of these orbit closures plays a central role in the infinite-dimensional representation theory of the real Lie group $U(p,q)$, and has applications to degeneracy loci and combinatorics. In this paper we use small resolutions to study orbit closures in this family. We prove that the fibers of these resolutions are smooth and strongly reduced, as well as a general result that if a variety has a resolution of singularities with these properties, then its characteristic cycle is irreducible. Hence these orbit closures have irreducible characteristic cycles. A result of Jones then allows us to calculate the torus-equivariant Chern-Mather classes of these orbit closures. We describe torus fixed points and tangent spaces of the resolutions, and use localization to obtain a formula for these classes. We conjecture that the Chern-Mather classes of a $K$-orbit closure are equivariantly positive when expressed in a Schubert basis of equivariant Borel-Moore homology, and use our results to verify the conjecture in an example.
We provide explicit formulas for computing the motivic Chern and Hirzebruch classes of degeneracy loci, especially those coming from the symplectic and odd orthogonal Grassmannians. The Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes arise as specializations of the motivic Chern and Hirzebruch classes. Our results are inspired by, and partially extends, those of Anderson--Chen--Tarasca in the case of ordinary Grassmannian degeneracy loci to isotropic and odd orthogonal Grassmannians as well as maximal even orthogonal Grassmannians. As applications, we obtain the motivic Chern and Hirzebruch classes of orthogonal and symplectic orbit closures in flag varieties.
We establish combinatorial and inductive formulas for Kazhdan-Lusztig polynomials associated to covexillary elements in classical types, extending results of Boe, Lascoux-Schützenberger, Sankaran-Vanchinathan, and Zelevinsky for Grassmannians of classical types. The proof uses intersection cohomology theory and the isomorphism of Kazhdan-Lusztig varieties from Anderson-Ikeda-Jeon-Kawago.
We establish two-pointed Prym-Brill-Noether loci with special vanishing at two points, and determine their K-theory classes when the dimensions are as expected. The classes are derived by the applications of a formula for the K-theory of certain vexillary degeneracy loci in type D. In particular, we show a two-pointed version of Prym-Petri theorem on the expected dimension in the general case, with a coupled Prym-Petri map. Our approach is inspired by the work on pointed cases by Tarasca, and we generalize unpointed cases by De Concini-Pragacz and Welters.
In this article we propose formulas for the connected K-theory class of the pointed Brill-Noether loci in Prym varieties, which extends the result by Concini and Pragacz. Applying the formulas, we compute the holomorphic Euler Characteristics of the loci.
We express a Schubert expansion of the Chern-Mather class for Schubert varieties in the even orthogonal Grassmannian via integrals involving Pfaffians and pushforward of the small resolutions in the sense of Intersection Cohomology (IH) constructed by Sankaran and Vanchinathan, instead of the Nash blowup. The equivariant localization is employed to show the way of computing the integral. As a byproduct, we present the computations. For analogy and the completion of the method in ordinary Grassmannians, we also suggest Kazhdan-Lusztig classes associated to Schubert varieties in the Lagrangian and odd orthogonal Grassmannian.
We use Jiang--Pareschi's cohomological rank functions and techniques developed by Caucci and Ito to study syzygies of projective bundles on abelian varieties. Consequently, we generalize and refine Chiantipalli's results regarding property $N_p$ of projective bundles over abelian varieties.
In a classical-type flag variety, we consider a Schubert variety associated to a vexillary (signed) permutation, and establish a combinatorial formula for the Hilbert-Samuel multiplicity of a point on such a Schubert variety. The formula is expressed in terms of excited Young diagrams, and extends results for Grassmannians due to Krattenthaler, Lakshmibai-Raghavan-Sankaran, and for the maximal isotropic (symplectic and orthogonal) Grassmannians to Ghorpade-Raghavan, Raghavan-Upadhyay, Kreiman, and Ikeda-Naruse. We also provide a new proof of a theorem of Li-Yong in the type A vexillary case. The main ingredient is an isomorphism between certain neighborhoods of fixed points, known as Kazhdan-Lusztig varieties, which, in turn, relies on a direct sum embedding previously used by Anderson-Fulton to relate vexillary loci to Grassmannian loci.