DSpace Community: KAIST Dept. of Mathematical Sciences
http://hdl.handle.net/10203/527
KAIST Dept. of Mathematical Sciences2021-09-03T09:34:22ZAlgebraic invariants of projections of varieties and partial elimination ideals
http://hdl.handle.net/10203/287513
Title: Algebraic invariants of projections of varieties and partial elimination ideals
Authors: Kwak, Sijong; Nguyen, Hop D.; Thanh Vu
Abstract: In this paper, we are interested in the properties of inner and outer projections with a view toward the Eisenbud-Goto regularity conjecture or the characterization of varieties satisfying certain extremal conditions. For example, if X is a quadratic scheme, the depth and regularity X and those of its inner projection from a smooth point are equal. In general, the above equalities do not hold for non-quadratic schemes. Therefore it is natural to investigate the algebraic invariants (e.g., depth and regularity) of X and its projected image in general. We develop a framework which provides partial answers and explains their relations using the partial elimination ideal theory. Our main theorems recover several preceding results in the literature. We also give some interesting examples and applications to illustrate our results. (C) 2021 Elsevier Inc. All rights reserved.2021-11-01T00:00:00ZDegree 3 unramified cohomology of classifying spaces for exceptional groups
http://hdl.handle.net/10203/285403
Title: Degree 3 unramified cohomology of classifying spaces for exceptional groups
Authors: Baek, Sanghoon
Abstract: Let G be a reductive group defined over an algebraically closed field of characteristic 0 such that the Dynkin diagram of G is the disjoint union of diagrams of types G(2), F-4, E-6, E-7, E-8. We show that the degree 3 unramified cohomology of the classifying space of G is trivial. In particular, combined with articles by Merkurjev [11] and the author [1], this completes the computations of degree 3 unramified cohomology and reductive invariants for all split semisimple groups of a homogeneous Dynkin type. (C) 2021 Elsevier B.V. All rights reserved.2021-10-01T00:00:00ZConformal Rigidity and Non-rigidity of the Scalar Curvature on Riemannian Manifolds
http://hdl.handle.net/10203/287549
Title: Conformal Rigidity and Non-rigidity of the Scalar Curvature on Riemannian Manifolds
Authors: Byeon, Jaeyoung; Jin, Sangdon
Abstract: For a compact smooth manifold (M, g(0)) with a boundary, we study the conformal rigidity and non-rigidity of the scalar curvature in the conformal class. It is known that the sign of the first eigenvalue for a linearized operator of the scalar curvature by a conformal change determines the rigidity/non-rigidity of the scalar curvature by conformal changes when the scalar curvature R-g0 is positive. In this paper, we show the sign condition of R-g0 is not necessary, and a reversed rigidity of the scalar curvature in the conformal class does not hold if there exists a point x(0) is an element of M with R-g0 (x(0)) > 0.2021-10-01T00:00:00ZComponentwise linearity of projective varieties with almost maximal degree
http://hdl.handle.net/10203/282869
Title: Componentwise linearity of projective varieties with almost maximal degree
Authors: Cuong, Doan Trung; Kwak, Sijong
Abstract: The degree of a projective subscheme has an upper bound deg(X) <= ((e+r)(e)) in terms of the codimension eand the reduction number r. It was proved in [3] that deg(X) = ((e r)(e)) if and only if Xis arithmetically Cohen-Macaulay and has an (r+ 1)-linear resolution. Moreover, if the degree of a projective variety Xsatisfies deg(X) = ((e+r)(e)) - 1, then the Betti table is described with some constraints. In this paper, we build on this work to show that most of such varieties are componentwise linear and the componentwise linearity is particularly suitable for understanding their Betti tables. As an application, the graded Betti numbers of those varieties with componentwise linear resolutions are computed. (C) 2021 Elsevier B.V. All rights reserved.2021-09-01T00:00:00Z