We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We consider the space
$X=\bigwedge ^{3}\mathbb{C}^{6}$
of alternating senary 3-tensors, equipped with the natural action of the group
$\operatorname{GL}_{6}$
of invertible linear transformations of
$\mathbb{C}^{6}$
. We describe explicitly the category of
$\operatorname{GL}_{6}$
-equivariant coherent
${\mathcal{D}}_{X}$
-modules as the category of representations of a quiver with relations, which has finite representation type. We give a construction of the six simple equivariant
${\mathcal{D}}_{X}$
-modules and give formulas for the characters of their underlying
$\operatorname{GL}_{6}$
-structures. We describe the (iterated) local cohomology groups with supports given by orbit closures, determining, in particular, the Lyubeznik numbers associated to the orbit closures.
We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is
$F$
-finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly
$F$
-regular varieties over arbitrary fields.
We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a three-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties
${\cal R}^i_k(A)$
.
We prove that the integral closure of a strongly Golod ideal in a polynomial ring over a field of characteristic zero is strongly Golod, positively answering a question of Huneke. More generally, the rational power
$I_{\unicode[STIX]{x1D6FC}}$
of an arbitrary homogeneous ideal is strongly Golod for
$\unicode[STIX]{x1D6FC}\geqslant 2$
and, if
$I$
is strongly Golod, then
$I_{\unicode[STIX]{x1D6FC}}$
is strongly Golod for
$\unicode[STIX]{x1D6FC}\geqslant 1$
. We also show that all the coefficient ideals of a strongly Golod ideal are strongly Golod.
Let (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.
In this paper, we prove that the set of all
$F$
-pure thresholds on a fixed germ of a strongly
$F$
-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all
$F$
-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.
We study
$F$
-signature under proper birational morphisms
$\unicode[STIX]{x1D70B}:Y\rightarrow X$
, showing that
$F$
-signature strictly increases for small morphisms or if
$K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$
. In certain cases, we can even show that the
$F$
-signature of
$Y$
is at least twice as that of
$X$
. We also provide examples of
$F$
-signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses.
Let R be a Mori domain with complete integral closure
$\widehat R$
, nonzero conductor
$\mathfrak f = (R: \widehat R)$
, and suppose that both v-class groups
${{\cal C}_v}(R)$
and
${{\cal C}_v}(3\widehat R)$
are finite. If
$R \mathfrak f$
is finite, then the elasticity of R is either rational or infinite. If
$R \mathfrak f$
is artinian, then unions of sets of lengths of R are almost arithmetical progressions with the same difference and global bound. We derive our results in the setting of v-noetherian monoids.
We develop the analog of crystalline Dieudonné theory for
$p$
-divisible groups in the arithmetic of function fields. In our theory
$p$
-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian
$t$
-modules and
$t$
-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.
We investigate whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a squarefree Borel ideal. We show that the ideal generated by the first multigraded shifts of a Borel ideal has linear quotients, as do the ideal generated by the
$k$
th multigraded shifts of a principal Borel ideal and an equigenerated squarefree Borel ideal for each
$k$
. Furthermore, we show that equigenerated squarefree Borel ideals share the property of being squarefree Borel with the ideals generated by multigraded shifts.
Fixing a positive integer r and
$0 \les k \les r-1$
, define
$f^{\langle r,k \rangle }$
for every formal power series f as
$ f(x) = f^{\langle r,0 \rangle }(x^r)+xf^{\langle r,1 \rangle }(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle }(x^r).$
Jochemko recently showed that the polynomial
$U^{n}_{r,k}\, h(x) := ( (1+x+\cdots +x^{r-1})^{n} h(x) )^{\langle r,k \rangle }$
has only non-positive zeros for any
$r \ges \deg h(x) -k$
and any positive integer n. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial
$h(x)$
of a lattice polytope of dimension n, which states that
$U^{n}_{r,0}\,h(x)$
has only negative, real zeros whenever
$r\ges n$
. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence
$( h^{\langle r,r-i \rangle }(x))_{1\les i \les r}$
is interlacing, so is
$( U^{n}_{r,r-i}\, h(x) )_{1\les i \les r}$
. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for coloured permutations. Besides, we derive a Carlitz identity for refined coloured permutations.
We prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring
$\mathscr{F}(E)$
of a finitely generated
$R$
-module
$E\subsetneq R^{e}$
over a Noetherian local ring
$R$
with infinite residue field. Assuming that
$R$
is Cohen–Macaulay of dimension 1 and that
$E$
has finite colength in
$R^{e}$
, our main result establishes an asymptotic length formula for the multiplicity of
$\mathscr{F}(E)$
, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of
$E$
in this setting. Further, we provide a Gorensteinness characterization for
$\mathscr{F}(E)$
in the more general situation where
$R$
is Cohen–Macaulay of arbitrary dimension and
$E$
is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module
$E$
if its reduction number is at least three.
We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.
We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring
$(R,\mathfrak{m},k)$
of prime characteristic to the real numbers at reduced parameter elements with respect to the
$\mathfrak{m}$
-adic topology.
In this paper, we will prove that any
$\mathbb{A}^{3}$
-form over a field
$k$
of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable
$\mathbb{A}^{2}$
-forms over a field
$k$
extends to
$\mathbb{A}^{2}$
-forms over any one-dimensional Noetherian domain containing
$\mathbb{Q}$
.
We study an operation, that we call lifting, creating nonisomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen–Macaulay tangent cones even if the tangent cone of the original curve is not Cohen–Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen–Macaulay tangent cones when the original monomial curve has a Cohen–Macaulay tangent cone. In this case, all the Betti sequences are just the Betti sequence of the original curve.
In this paper, we study the singularities of a general hyperplane section
$H$
of a three-dimensional quasi-projective variety
$X$
over an algebraically closed field of characteristic
$p>0$
. We prove that if
$X$
has only canonical singularities, then
$H$
has only rational double points. We also prove, under the assumption that
$p>5$
, that if
$X$
has only klt singularities, then so does
$H$
.
Let
$R$
be a commutative Noetherian ring of prime characteristic
$p$
. In this paper, we give a short proof using filter regular sequences that the set of associated prime ideals of
$H_{I}^{t}(R)$
is finite for any ideal
$I$
and for any
$t\geqslant 0$
when
$R$
has finite
$F$
-representation type or finite singular locus. This extends a previous result by Takagi–Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of
$R$
(in any characteristic) to guarantee that the set
$\operatorname{Ass}H_{I}^{2}(R)$
is always finite.
Given a nonincreasing function f : ℤ≥ 0 \{0} → ℤ≥ 0 such that (i) f(k) − f(k + 1) ≤ 1 for all k ≥ 1 and (ii) if a = f(1) and b = limk → ∞f(k), then |f−1(a)| ≤ |f−1(a − 1)| ≤ ··· ≤ |f−1(b + 1)|, a system of generators of a monomial ideal I ⊂ K[x1, . . ., xn] for which depth S/Ik = f(k) for all k ≥ 1 is explicitly described. Furthermore, we give a characterization of triplets of integers (n, d, r) with n > 0, d ≥ 0 and r > 0 with the properties that there exists a monomial ideal I ⊂ S = K[x1, . . ., xn] for which limk→∞ depth S/Ik = d and dstab(I) = r, where dstab(I) is the smallest integer k0 ≥ 1 with depth S/Ik0 = depth S/Ik0+1 = depth S/Ik0+2 = ···.
This paper is a complement to the work of the second author on modular quotient singularities in odd characteristic. Here, we prove that if V is a three-dimensional vector space over a field of characteristic 2 and G < GL(V) is a finite subgroup generated by pseudoreflections and possessing a two-dimensional invariant subspace W such that the restriction of G to W is isomorphic to the group SL2(𝔽2n), then the quotient V/G is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities that are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.