Jet Support Closure in Algebraic Geometry

• Jet support closure is a closure operation on ideals defined using reduced local jet schemes, capturing only the support of local jet fibers.
• It bridges jet closure and integral closure by recovering the latter in regular rings and exhibiting finer behavior in singular settings.
• The operation serves as an invariant of singularities and is computable for monomial, homogeneous, and simple singularities, aiding in local isomorphism studies.

Searching arXiv for papers directly relevant to jet support closure in algebraic geometry. Jet support closure is a closure operation on ideals defined from reduced local jet schemes. In the framework introduced to study the local isomorphism problem, it assigns to an ideal $a\subset R$ the largest ideal cutting out the same reduced local $m$-jet scheme as $a$, and, after passage to all orders, a support-type analogue of arc closure. The notion sits between jet closure and integral closure: it is weaker than jet closure, recovers ordinary integral closure in regular local rings essentially of finite type over $k$, and in singular rings can be strictly smaller. Later work recast the quotients $R/I^{m\text{-jsc}}$ as invariants of singularities and computed them in homogeneous, monomial, weighted homogeneous, and simple-singularity settings (Fernex et al., 2017); (Chen et al., 8 Jul 2025).

1. Origin in the local isomorphism problem

The original setting is a morphism of germs of $k$-schemes

$\varphi:(Y,y)\to (X,x),$

which induces, for each $m\in \mathbb N\cup\{\infty\}$, a morphism on local $m$-jet schemes

$\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$

The basic question asks whether isomorphisms on all local jet schemes force $m$0 itself to be an isomorphism. This is the local isomorphism problem. Its embedded version asks the same question under the additional assumption that $m$1 is a closed immersion. The closure operations introduced from jet schemes were designed precisely to translate these geometric questions into ideal-theoretic ones (Fernex et al., 2017).

For a scheme $m$2, the $m$3-jet scheme $m$4 represents morphisms

$m$5

while the arc space $m$6 is the inverse limit of the $m$7, representing maps from $m$8. For a point $m$9, the local $a$0-jets are the fiber

$a$1

Jet support closure arises by asking not for equality of local jet schemes as schemes, but only for equality after reduction. In that sense it is a support-theoretic weakening of jet closure, adapted to the geometry of reduced local jet fibers (Fernex et al., 2017).

2. Definition through local jet fibers

Let $a$2 be a local $a$3-algebra, let $a$4, and let $a$5 be the closed point. For an ideal $a$6, the associated ideal $a$7 is defined using Hasse–Schmidt derivations by

$a$8

The $a$9-jet closure and the $k$0-jet support closure are then defined in parallel (Fernex et al., 2017).

Operation	Definition	Geometric characterization
$k$1-jet closure $k$2	$k$3 with $k$4	largest $k$5 with $k$6
$k$7-jet support closure $k$8	$k$9 with $R/I^{m\text{-jsc}}$0	largest $R/I^{m\text{-jsc}}$1 with $R/I^{m\text{-jsc}}$2
Infinite-order versions	$R/I^{m\text{-jsc}}$3, $R/I^{m\text{-jsc}}$4	arc closure and arc support closure

The defining distinction is reduction: jet closure remembers the full scheme structure of the local jet fiber, whereas jet support closure remembers only its support. Equivalently, $R/I^{m\text{-jsc}}$5 is the largest ideal defining a subscheme with the same reduced local $R/I^{m\text{-jsc}}$6-jets as $R/I^{m\text{-jsc}}$7. This is the precise sense in which jet support closure is weaker than jet closure but still controlled by local jet geometry (Fernex et al., 2017).

Later work gives a closely related formulation

$R/I^{m\text{-jsc}}$8

and expresses jet support closure as a kernel of a natural jet-theoretic map. In particular, for the zero ideal there is a map

$R/I^{m\text{-jsc}}$9

with

$k$0

This formulation makes the support-theoretic character of the construction particularly explicit (Chen et al., 8 Jul 2025).

3. Structural properties and closure hierarchy

Jet support closure is part of a small hierarchy of jet-theoretic closure operations. For every ideal $k$1,

$k$2

Passing to all finite orders, the paper defines

$k$3

and shows that this is itself a closure operation: $k$4 It also establishes the chain

$k$5

where $k$6 is the arc closure and $k$7 is the arc support closure. Thus $k$8 is intermediate between the scheme-theoretic arc closure and the support-theoretic infinite-order closure (Fernex et al., 2017).

From the later invariant-theoretic viewpoint, jet support closure is compatible with quotient formation. If $k$9 is the quotient map, then jet support closure of $\varphi:(Y,y)\to (X,x),$0 in $\varphi:(Y,y)\to (X,x),$1 agrees with jet support closure of the zero ideal in $\varphi:(Y,y)\to (X,x),$2. Moreover, if $\varphi:(Y,y)\to (X,x),$3 is an isomorphism and $\varphi:(Y,y)\to (X,x),$4, then

$\varphi:(Y,y)\to (X,x),$5

for every $\varphi:(Y,y)\to (X,x),$6. In this sense the algebras $\varphi:(Y,y)\to (X,x),$7 are invariants of the singularity $\varphi:(Y,y)\to (X,x),$8, rather than artifacts of a chosen embedding (Chen et al., 8 Jul 2025).

The support closure is therefore both geometric and intrinsic. Geometrically it is characterized by reduced local jet fibers; algebraically it is idempotent and functorial enough to descend to singularity invariants. This dual character explains why the notion is useful both in the local isomorphism problem and in explicit singularity calculations.

4. Comparison with integral closure

A central comparison theorem places jet support closure against classical integral closure. If $\varphi:(Y,y)\to (X,x),$9 is a local integral domain essentially of finite type over $m\in \mathbb N\cup\{\infty\}$0, then for every ideal $m\in \mathbb N\cup\{\infty\}$1,

$m\in \mathbb N\cup\{\infty\}$2

where $m\in \mathbb N\cup\{\infty\}$3 denotes the integral closure. If, in addition, $m\in \mathbb N\cup\{\infty\}$4 is regular, then equality holds: $m\in \mathbb N\cup\{\infty\}$5 Accordingly, in regular local rings jet support closure recovers ordinary integral closure, while in singular rings it can be strictly smaller and hence more sensitive (Fernex et al., 2017).

The standard example in the singular case is

$m\in \mathbb N\cup\{\infty\}$6

Here

$m\in \mathbb N\cup\{\infty\}$7

but $m\in \mathbb N\cup\{\infty\}$8 is nevertheless jet support closed. This example shows that jet support closure is not merely a reformulation of integral closure. It detects a finer relationship between an ideal and the reduced local jet fibers of its zero set (Fernex et al., 2017).

Later computations reinforce that finer behavior. For monomial ideals, jet support closure remains monomial, but it does not behave additively in general: $m\in \mathbb N\cup\{\infty\}$9 That failure is consistent with its geometric definition: equality of reduced local jet fibers is not additive in the same way as many valuation-theoretic constructions (Chen et al., 8 Jul 2025).

5. Relation to the embedded local isomorphism problem

The closure-theoretic approach achieves its strongest geometric consequence through arc closure. For a germ $m$0 with local ring $m$1, the germ has the embedded local isomorphism property if and only if the zero ideal is arc closed: $m$2 Since

$m$3

the embedded local isomorphism problem becomes the question whether any nonzero function can have the same local arcs as $m$4. Jet support closure enters this framework as the weaker support-type companion to jet closure, clarifying what remains visible after reduction of the local jet schemes (Fernex et al., 2017).

The paper proves that the embedded local isomorphism problem has a negative answer in general. There exists a local $m$5-algebra whose zero ideal is not arc closed; the construction proceeds from

$m$6

with infinitely many variables and a suitable ideal $m$7, and passing to $m$8 yields a local algebra whose zero ideal is not arc closed. Consequently, there exist germs that do not have the embedded local isomorphism property (Fernex et al., 2017).

At the same time, several positive cases are established. The zero ideal is arc closed if $m$9 is a graded local $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$0-algebra, if $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$1 is a reduced Noetherian local algebra essentially of finite type over $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$2, or if $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$3 where $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$4 is regular essentially of finite type over $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$5 and $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$6. Hence the embedded local isomorphism property holds for homogeneous germs, germs of reduced schemes of finite type over $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$7, and germs of hypersurfaces in smooth varieties. The paper also leaves open the Noetherian question: if $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$8 is a Noetherian local $\varphi_m^{\mathrm{loc}}:Y_y^m\to X_x^m.$9-algebra, is every ideal arc closed? A positive answer would extend the embedded local isomorphism property to all Noetherian germs, while a negative answer would exhibit a nontrivial closure operation stronger than integral closure in the Noetherian setting (Fernex et al., 2017).

6. Later invariant-theoretic and computational developments

Later work develops jet support closure as an explicit invariant of singularities. It introduces two local algebras associated to jet closure and jet support closure, proves that they are invariants under analytic isomorphism, and investigates them for monomial ideals, homogeneous ideals, and simple plane curve singularities. In that framework, jet support closure becomes not only a conceptual bridge from jet geometry to local algebra, but also a computable object (Chen et al., 8 Jul 2025).

Several classes admit especially simple formulas. If $m$00 is a monomial ideal with square-free generators, then

$m$01

For a reduced homogeneous polynomial $m$02,

$m$03

In these cases jet closure and jet support closure coincide with the obvious truncation by $m$04, so the reduced local jet fiber retains exactly the expected finite-order information (Chen et al., 8 Jul 2025).

For general monomial ideals, the description is combinatorial. A monomial

$m$05

lies in $m$06 if and only if for every $m$07 satisfying

$m$08

there exists a monomial generator

$m$09

such that

$m$10

For weighted homogeneous polynomials in two variables the paper gives explicit piecewise formulas. For instance, for the $m$11 singularity $m$12, if $m$13, then

$m$14

Analogous formulas are obtained for $m$15, $m$16, and $m$17 (Chen et al., 8 Jul 2025).

These explicit computations feed back into the local isomorphism problem. For simple curve singularities, if

$m$18

then

$m$19

Thus finitely many jet support closures determine the singularity in this class. The same paper also introduces a jet filtration and a jet index, primarily for jet closure, thereby extending the original closure-theoretic program from existence questions to quantitative invariants of how finite jet orders recover the base scheme (Chen et al., 8 Jul 2025).