In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal [Formula: see text] generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal [Formula: see text], and describe the minimal graded free resolution of the symmetric algebra of [Formula: see text]. Finally, we provide a method to compute the defining equations of the Rees algebra of [Formula: see text] using three moving planes that follow the parametrization.
Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of $D$-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some $D$-module theoretic invariant given by the degree of the Kashiwara operator.
Let [Formula: see text] be a commutative ring with unity, [Formula: see text] and [Formula: see text] an ideal of [Formula: see text]. Define [Formula: see text] to be [Formula: see text] a quotient of the Rees algebra. In this paper, we investigate when the rings in the family are generalized Cohen–Macaulay or filter rings and show that these properties are independent of the choice of [Formula: see text] and [Formula: see text].
Let
R
R
be the face ring of a simplicial complex of dimension
d
−
1
d-1
and
R
(
n
)
{\mathcal R}({\mathfrak {n}})
be the Rees algebra of the maximal homogeneous ideal
n
{\mathfrak {n}}
of
R
.
R.
We show that the generalized Hilbert-Kunz function
H
K
(
s
)
=
ℓ
(
R
(
n
)
/
(
n
,
n
t
)
[
s
]
)
HK(s)=\ell ({\mathcal {R}}({\mathfrak {n}})/({\mathfrak {n}}, {\mathfrak {n}} t)^{[s]})
is given by a polynomial for all large
s
.
s.
We calculate it in many examples and also provide a Macaulay2 code for computing
H
K
(
s
)
.
HK(s).
Abstract
The regularity $${\text {reg}}R(I(G))$$
reg
R
(
I
(
G
)
)
of the Rees ring R(I(G)) of the edge ideal I(G) of a finite simple graph G is studied. It is shown that, if R(I(G)) is normal, one has $${\text {mat}}(G) \le {\text {reg}}R(I(G)) \le {\text {mat}}(G) + 1$$
mat
(
G
)
≤
reg
R
(
I
(
G
)
)
≤
mat
(
G
)
+
1
, where $${\text {mat}}(G)$$
mat
(
G
)
is the matching number of G. In general, the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.
The Hilbert-Kunz function of some quadratic quotients of the Rees algebra