Papers by Tran Nguyen Khanh Linh
arXiv (Cornell University), Mar 24, 2020
Given a fat point scheme W = m 1 P 1 + • • • + m s P s in the projective n-space P n over a field... more Given a fat point scheme W = m 1 P 1 + • • • + m s P s in the projective n-space P n over a field K of characteristic zero, the modules of Kähler differential k-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of W when k ∈ {1, . . . , n + 1}. In this paper we determine the value of its Hilbert polynomial explicitly for the case k = n + 1, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme Y = (m 1 -1)P 1 +• • •+(m s -1)P s . For n = 2, this allows us to determine the Hilbert polynomials of the modules of Kähler differential k-forms for k = 1, 2, 3, and to produce a sharp bound for the regularity index for k = 2.
arXiv (Cornell University), Jul 5, 2021
Given an ACM set X of points in a multiprojective space P m × P n over a field of characteristic ... more Given an ACM set X of points in a multiprojective space P m × P n over a field of characteristic zero, we are interested in studying the Kähler different and the Cayley-Bacharach property for X. In P 1 × P 1 , the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the Kähler different. However, this result fails to hold in P m × P n for n > 1 or m > 1. In this paper we start an investigation of the Kähler different and its Hilbert function and then prove that X is a complete intersection of type (d 1 , ..., dm, d 1 , ..., d n) if and only if it has the Cayley-Bachrach property and the Kähler different is non-zero at a certain degree. When X has the ()-property, we characterize the Cayley-Bacharach property of X in terms of its components under the canonical projections.
arXiv (Cornell University), Nov 29, 2016
Let X be a set of K-rational points in P 1 × P 1 over a field K of characteristic zero, let Y be ... more Let X be a set of K-rational points in P 1 × P 1 over a field K of characteristic zero, let Y be a fat point scheme supported at X, and let R Y be the bihomogeneus coordinate ring of Y. In this paper we investigate the module of Kähler differentials Ω 1 R Y /K. We describe this bigraded R Y-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support X is a complete intersection or an almost complete intersection in P 1 × P 1. Moreover, we introduce a Kähler different for Y and use it to characterize ACM reduced schemes in P 1 × P 1 having the Cayley-Bacharach property.
arXiv (Cornell University), Apr 12, 2017
Given a 0-dimensional scheme X in a projective space P n K over a field K, we characterize the Ca... more Given a 0-dimensional scheme X in a projective space P n K over a field K, we characterize the Cayley-Bacharach property of X in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley-Bacharach schemes by Dedekind's formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes.
Taiwanese Journal of Mathematics, Jun 1, 2020
Given a 0-dimensional scheme X in a n-dimensional projective space P n K over an arbitrary field ... more Given a 0-dimensional scheme X in a n-dimensional projective space P n K over an arbitrary field K, we use Liaison theory to characterize the Cayley-Bacharach property of X. Our result extends the result for sets of Krational points given in [7]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of X when X has the Cayley-Bacharach property. Theorem 1.1. Let W be a set of points in P n K which is a complete intersection, let X ⊆ W, let Y = W \ X, and let I W , I X and I Y denote the homogeneous vanishing ideals of W, X and Y in P = K[X 0 , ..., X n ], respectively. Set α Y/W = min{i ∈ N | (I Y /I W) i = 0 }. Then the following conditions are equivalent. (a) X is a Cayley-Bachrach scheme. (b) A generic element of (I Y) α Y/W does not vanish at any point of X. (c) We have I W : (I Y) α Y/W = I X .
Acta Mathematica Vietnamica, 2021
Given a fat point scheme W = m 1 P 1 + • • • + m s P s in the projective n-space P n over a field... more Given a fat point scheme W = m 1 P 1 + • • • + m s P s in the projective n-space P n over a field K of characteristic zero, the modules of Kähler differential k-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of W when k ∈ {1,. .. , n + 1}. In this paper we determine the value of its Hilbert polynomial explicitly for the case k = n + 1, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme Y = (m 1 −1)P 1 +• • •+(m s −1)P s. For n = 2, this allows us to determine the Hilbert polynomials of the modules of Kähler differential k-forms for k = 1, 2, 3, and to produce a sharp bound for the regularity index for k = 2.
arXiv (Cornell University), Apr 7, 2017
Given a 0-dimensional scheme in a projective n-space P n over a field K, we study the Kähler diff... more Given a 0-dimensional scheme in a projective n-space P n over a field K, we study the Kähler differential algebra Ω R X /K of its homogeneous coordinate ring R X. Using explicit presentations of the modules Ω m R X /K of Kähler differential m-forms, we determine many values of their Hilbert functions explicitly and bound their Hilbert polynomials and regularity indices. Detailed results are obtained for subschemes of P 1 , fat point schemes, and subschemes of P 2 supported on a conic.
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Papers by Tran Nguyen Khanh Linh