Abstract
Let 핐 be a fat point scheme in ℙn1 × … × ℙnk over a field K of characteristic zero. In this paper we introduce the multi-graded Kähler differential module for 핐 and we establish a short exact sequence of this module in terms of the thickening of 핐.
Highlights
In [1], G. de Dominicis and M
Given a finite set of points X in Pn with homogeneous vanishing ideal IX in R = K[X0, . . . , Xn] and homogeneous coordinate ring RX = R/IX, the Kähler differential module for X is the RX-module Ω1RX/K = J/J2 where J is the kernel of the multiplication map μ : RX ⊗K RX → RX
0 → IX(2)/IX2 → IX/IX2 → RnX+1(−1) → Ω1RX/K → 0. Based on this exact sequence, the structure of this module can be precisely described in several special cases, for instance, if X is the complete intersection of hypersurfaces of degrees d1, . . . , dn (n + 1) HFX(i −
Summary
In [1], G. de Dominicis and M. In this paper we will consider the natural question of whether these differential algebraic methods can be applied to study fat point schemes of a multiprojective space Pn1 × · · · × Pnk and, especially, we look closely at the generalization of the above canonical exact sequence to fat point schemes in Pn1 × · · · × Pnk. Let Y be a fat point scheme in Pn1 × · · · × Pnk with multihomogeneous vanishing ideal IY ⊆ S = K[X10, . 0 −→ IY/IV −→ RnY1+1(−e1) ⊕ · · · ⊕ RYnk+1(−ek) −→ Ω1RY/K −→ 0 This result shows that one can compute the Hilbert function of the Kähler differential module for Y from the Hilbert functions of Y and V, in to compute the Hilbert function HFΩ1RY/K (i), we need to compute HFΩ1RY/K (i) for only a finite number of i ∈ Zk
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