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1. CJM Online first
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$ |
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$ We exhibit a set of minimal generators of the defining ideal of the
Rees Algebra associated with the ideal of three bivariate homogeneous
polynomials parametrizing a proper rational curve in projective plane,
having a minimal syzygy of degree 2.
Keywords:Rees Algebras, rational plane curves, minimal generators Categories:13A30, 14H50 |
2. CJM 2009 (vol 61 pp. 762)
The Hilbert Coefficients of the Fiber Cone and the $a$-Invariant of the Associated Graded Ring Let $(A,\m)$ be a Noetherian local ring with infinite residue
field and let $I$ be an ideal in $A$ and let $F(I) =
\bigoplus_{n \geq 0}I^n/\m I^n$ be the fiber cone of $I$.
We prove certain relations among the Hilbert coefficients $f_0(I),f_1(I), f_2(I)$ of $F(I)$
when the $a$-invariant of the associated graded ring $G(I)$ is negative.
Keywords:fiber cone, $a$-invariant, Hilbert coefficients of fiber cone Categories:13A30, 13D40 |
3. CJM 2007 (vol 59 pp. 109)
On Fiber Cones of $\m$-Primary Ideals Two formulas for the multiplicity of the fiber cone
$F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of
a $d$-dimensional Cohen--Macaulay local ring $(R,\m)$ are derived in
terms of the mixed multiplicity $e_{d-1}(\m | I)$, the multiplicity
$e(I)$, and superficial elements. As a consequence, the
Cohen--Macaulay property of $F(I)$ when $I$ has minimal mixed
multiplicity or almost minimal mixed multiplicity is characterized
in terms of the reduction number of $I$ and lengths of certain ideals.
We also characterize the Cohen--Macaulay and Gorenstein properties of
fiber cones of $\m$-primary ideals with a $d$-generated minimal
reduction $J$ satisfying $\ell(I^2/JI)=1$ or
$\ell(I\m/J\m)=1.$
Keywords:fiber cones, mixed multiplicities, joint reductions, Cohen--Macaulay fiber cones, Gorenstein fiber cones, ideals having minimal and almost minimal mixed multiplicities Categories:13H10, 13H15, 13A30, 13C15, 13A02 |