Jet Closure in Algebraic Geometry

• Jet closure is a closure operation on ideals in local k-algebras defined via local m-jet schemes, distinguishing ideals by their m-th order infinitesimal behavior.
• It plays a critical role in studying singularities and resolving the local isomorphism problem, with variants like arc closure and jet support closure offering nuanced insights.
• Explicit calculations in regular and singular rings show that jet closure can capture higher-order conditions that ordinary ideal membership may miss.

Searching arXiv for papers specifically on “jet closure” in algebraic geometry and commutative algebra. Jet closure is a closure operation on ideals of a local $k$-algebra defined through the behavior of local jet schemes. Introduced by de Fernex, Ein, and Ishii in the study of the local isomorphism problem, it formalizes when two ideals are indistinguishable by $m$-th order infinitesimal data at a point; its limiting version is arc closure, defined through all finite jet levels at once (Fernex et al., 2017). Subsequent work has recast jet closure and jet support closure as invariants of singularities, introduced associated local algebras and filtrations, and computed them explicitly for homogeneous ideals, monomial ideals, and simple plane curve singularities (Chen et al., 8 Jul 2025).

1. Jet schemes, local jets, and the definition of jet closure

For a scheme $X$ over a field $k$, the $m$-jet scheme $X_m$ represents the functor

$$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$

and the arc space $X_\infty$ is the inverse limit $\varprojlim_m X_m$ (Fernex et al., 2017). If $x\in X$, the local $m$0-jets at $m$1 are the fiber

$m$2

where $m$3 is the canonical projection.

In the affine case $m$4, the jet rings $m$5 are described using universal Hasse–Schmidt derivations $m$6, or equivalently the Hasse–Schmidt algebra $m$7 when $m$8 is a $m$9-algebra (Fernex et al., 2017, Chen et al., 8 Jul 2025). For an ideal $X$0, its $X$1-th jet ideal is

$X$2

For a local $X$3-algebra $X$4, the $X$5-jet closure of $X$6 is

$X$7

The case $X$8 is the arc closure,

$X$9

Geometrically, $k$0 is the largest ideal $k$1 such that the local $k$2-jet schemes of $k$3 and $k$4 coincide at the closed point (Fernex et al., 2017).

2. Variants, intrinsic structure, and relation to integral closure

Jet closure sits inside a small family of jet-theoretic closure operations (Fernex et al., 2017).

Operation	Definition	Geometric criterion
$k$5-jet closure $k$6	$k$7	equality of local $k$8-jet schemes
arc closure $k$9	$m$0	equality at all finite jet levels
$m$1-jet support closure $m$2	$m$3	equality of reduced local $m$4-jet schemes
jet support closure $m$5	$m$6	equality of reduced local jets for all finite $m$7
arc support closure $m$8	$m$9	reduced equality at arc level

These operations are intrinsic in the sense that the $X_m$0-jet closure of $X_m$1 is the inverse image of the $X_m$2-jet closure of the zero ideal in $X_m$3, and the same holds for the support variants (Fernex et al., 2017). For the zero ideal, jet closure admits a kernel description: if $X_m$4 is induced by the universal local $X_m$5-jet, then

$X_m$6

Several formal properties hold. Jet closure and jet support closure are extensive and idempotent, and jet closure satisfies

$X_m$7

Arc closure is the intersection of finite jet closures,

$X_m$8

and there are containments

$X_m$9

Moreover,

$$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$0

for every $$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$1 (Fernex et al., 2017).

The main comparison with classical closure theory is through jet support closure. If $$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$2 is a local integral domain essentially of finite type over $$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$3, then

$$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$4

and if $$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$5 is regular, then

$$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$6

so in regular local rings essentially of finite type, jet support closure recovers integral closure (Fernex et al., 2017).

3. The local isomorphism problem

The original motivation for jet closure was the local isomorphism problem: if a morphism of germs

$$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$7

induces isomorphisms on all local jet schemes

$$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$8

for every $$Z \mapsto \operatorname{Hom}_k\!\bigl(Z\times_k \operatorname{Spec} k[t]/(t^{m+1}),X\bigr),$$9, must $X_\infty$0 be an isomorphism of germs? The embedded version asks the same question under the additional assumption that $X_\infty$1 is a closed immersion (Fernex et al., 2017).

For a local algebra $X_\infty$2, the embedded local isomorphism property is equivalent to the arc closedness of the zero ideal: $X_\infty$3 Thus the problem becomes algebraic: whether nonzero functions can vanish on all local arcs through the closed point (Fernex et al., 2017).

The answer is negative in full generality. In the non-Noetherian ring

$X_\infty$4

one has $X_\infty$5 but $X_\infty$6 for every $X_\infty$7, hence $X_\infty$8 (Fernex et al., 2017).

Positive results were obtained for several important classes. The zero ideal is arc closed when $X_\infty$9 is a graded local $\varprojlim_m X_m$0-algebra, when $\varprojlim_m X_m$1 is a reduced Noetherian local $\varprojlim_m X_m$2-algebra essentially of finite type over $\varprojlim_m X_m$3, and when $\varprojlim_m X_m$4 with $\varprojlim_m X_m$5 regular essentially of finite type over $\varprojlim_m X_m$6 (Fernex et al., 2017). Later, Mallory’s theorem established that in Noetherian local $\varprojlim_m X_m$7-algebras with separable residue field, every ideal is arc closed; as recorded in later work, this gives embedded local isomorphism for Noetherian test germs in that setting (Chen et al., 8 Jul 2025).

4. Explicit computations and basic examples

Jet closure is nontrivial even in regular complete local rings. In $\varprojlim_m X_m$8, with $\varprojlim_m X_m$9 and $x\in X$0,

$x\in X$1

showing that finite jet closure records higher-order conditions beyond ordinary ideal membership (Fernex et al., 2017).

Jet support closure can be strictly smaller than integral closure in singular rings. In

$x\in X$2

the element $x\in X$3 is integral over $x\in X$4, so $x\in X$5, but $x\in X$6 is jet support closed, hence

$x\in X$7

(Fernex et al., 2017).

Recent work gives large computable classes. If $x\in X$8 and $x\in X$9 is extended from a homogeneous ideal in the polynomial ring, then for every $m$00,

$m$01

For a principal homogeneous ideal $m$02 with $m$03, this yields

$m$04

with the stated convention for negative binomial arguments (Chen et al., 8 Jul 2025).

For monomial ideals, jet support closure is again monomial, and for square-free monomial generators both jet closure and jet support closure satisfy

$m$05

for all $m$06 (Chen et al., 8 Jul 2025). By contrast, in $m$07 the homogeneous ideal

$m$08

satisfies

$m$09

so jet support closure is more sensitive than the homogeneous formula for jet closure (Chen et al., 8 Jul 2025).

5. Singularities, associated local algebras, and the jet index

A central development is the introduction of the local algebras

$m$10

for $m$11. These are invariants of the singularity germ: if $m$12 as local rings, then

$m$13

for every $m$14 (Chen et al., 8 Jul 2025).

The $m$15-jet closures form a descending chain

$m$16

with

$m$17

This yields a filtration $m$18 on $m$19 in the sense of Rees; it satisfies

$m$20

and one also has

$m$21

(Chen et al., 8 Jul 2025).

When $m$22 is Artinian, the descending chain stabilizes: there exists $m$23 such that

$m$24

The smallest such $m$25 is the jet index $m$26, which measures the finite jet level at which jet closure recovers the ideal (Chen et al., 8 Jul 2025).

For isolated hypersurface singularities, this produces explicit invariants. If

$m$27

then for the Jacobian ideal $m$28,

$m$29

(Chen et al., 8 Jul 2025). The same work defines jet Milnor and jet Tjurina indices using $m$30 and $m$31, and formulates the conjecture

$m$32

for isolated singularities.

The simple plane curve singularities $m$33 exhibit especially explicit behavior. For example, if

$m$34

then

$m$35

while at $m$36,

$m$37

More generally, later work shows that simple plane curve singularities are classified by finitely many jet support closure algebras: if $m$38 is the maximum of the Milnor numbers of two simple singularities, then

$m$39

(Chen et al., 8 Jul 2025).

6. Terminology and cross-disciplinary usage

In algebraic geometry and commutative algebra, jet closure refers to the closure operations just described, built from local jet schemes, Hasse–Schmidt derivations, and arc spaces (Fernex et al., 2017, Chen et al., 8 Jul 2025). The term is not uniform across the wider arXiv literature. In fluid mechanics, “closure” in jet problems refers to resonance closure in jet screech or turbulence closure in free and plunging jets (Nogueira et al., 2021, Li et al., 2023, Azim, 2021). In relativistic astrophysics, “closure relations” for structured jets link temporal and spectral slopes in gamma-ray burst afterglows (Ryan et al., 2019). This disciplinary divergence is substantive rather than terminological: only the algebraic-geometric usage defines a closure operation on ideals through jet schemes.

Within its own field, jet closure occupies a distinctive position among closure operations. It is defined by infinitesimal test objects rather than by valuations, Frobenius powers, or continuous coefficients, and jet support closure interfaces directly with integral closure in regular local rings (Fernex et al., 2017). The subsequent introduction of the algebras $m$40, $m$41, the filtration $m$42, and the jet index shows that jet closure is not only a device for the local isomorphism problem, but also a source of computable invariants of singularities (Chen et al., 8 Jul 2025).