Segregativity Criterion in Algebraic Geometry

• Segregativity Criterion is a theorem stating that the integral dependence of ideal sheaves is equivalent to equality of Segre classes in an equidimensional projective scheme.
• It provides a global, intersection-theoretic measure linking birational invariance of Segre classes to precise numerical and Chow class equivalences.
• The criterion extends to homogeneous ideals via Aluffi’s Segre zeta function, offering a complete invariant for determining integral closure.

Searching arXiv for the exact topic and the primary cited paper to ground the article. Searching arXiv for "Segre classes and integral dependence" and "Segregativity Criterion". The Segregativity Criterion is the theorem that, for closed subschemes of an equidimensional projective scheme, integral dependence of ideal sheaves is equivalent to equality of Segre classes, and is even equivalent to equality of all Segre-class degrees with respect to any ample line bundle. In the formulation given in "Segre classes and integral dependence" (Cid-Ruiz, 9 Dec 2025), the criterion reverses the classical implication coming from birational invariance: Segre classes not only depend on integral closure, but also detect it.

1. Definition and geometric setting

Let $X$ be an equidimensional projective scheme over a field $k$, and let $L$ be an ample line bundle on $X$. Let $W \subseteq Z \subseteq X$ be two closed subschemes with ideal sheaves

$I_W \subseteq I_Z \subseteq \mathcal{O}_X .$

In this setting, the Segregativity Criterion is the equivalence

$$I_W \text{ is integral over } I_Z \iff s(Z, X) = s(W, X),$$

together with equivalent numerical formulations in terms of degrees of Segre-class components (Cid-Ruiz, 9 Dec 2025).

The criterion is stated for Segre classes $s(Z,X)$ of closed subschemes. The Segre class is given by

$$s(Z, X) = q_* \left( \sum_{i \ge 0} c_1(\mathcal{O}_{(C \oplus 1)}(1))^i \frown [(C \oplus 1)] \right ),$$

where $C = C_Z X$ is the normal cone of $k$0 in $k$1, and $k$2 is projection to $k$3.

In this formulation, the criterion is global and intersection-theoretic. It concerns ideal sheaves on projective schemes rather than only local algebra, and it packages integral dependence into Chow-theoretic data.

2. Birational invariance and the converse statement

A fundamental property of Segre classes is birational invariance. This invariance implies that the Segre class of a closed subscheme only depends on the integral closure of the defining ideal sheaf. Formally,

$k$4

in the appropriate Chow group (Cid-Ruiz, 9 Dec 2025).

The main point of the Segregativity Criterion is that this relationship is bidirectional. Equality of Segre classes does not merely follow from equality of integral closures; it characterizes integral dependence. In the language of the paper, equality of Segre classes characterizes integral dependence of ideals.

This reverses the usual direction of use. Rather than treating Segre classes as invariants insensitive to integral closure, the criterion shows that they encode enough information to recover an integral dependence test. This suggests that the Segre class is not only birationally stable data, but also a complete detector for the integral-closure relation in the stated projective setting.

3. Equivalent formulations of the Segregativity Criterion

The main theorem, identified in the source as Theorem A / Theorem 4.4, states that the following conditions are equivalent (Cid-Ruiz, 9 Dec 2025):

Criterion	Statement
Integral dependence	$k$5 is integral over $k$6
Chow class equality	$k$7 in $k$8
Numerical equality	$k$9 for all $L$0

Equivalently,

$L$1

The theorem also includes further equivalent conditions: equality of the degrees of all components of the Vogel cycles associated to $L$2 and $L$3, and equality of degrees of certain polar cycles constructed from the Segre data.

The presence of these additional formulations is significant because it places the criterion simultaneously in three languages: ideal-theoretic, Chow-theoretic, and numerical. In particular, the numerical form shows that one does not need the full class equality as input; equality of all degrees with respect to an ample polarization already suffices.

4. Numerical testing and the necessity of ampleness

The numerical form of the criterion depends essentially on the hypothesis that $L$4 is ample. The source states that if $L$5 is only big and nef, the criterion can fail: two subschemes can have distinct Segre classes yet the intersection numbers with respect to a big and nef $L$6 vanish and thus coincide (Cid-Ruiz, 9 Dec 2025).

The explicit example is as follows. Let $L$7 be the blow-up of $L$8 first at a point, then along a line in the exceptional divisor. Let $L$9 be the exceptional divisor of the second blowup, and let $X$0 be a multiple Cartier divisor along $X$1, with $X$2. Let

$X$3

which is big and nef, not ample. Then

$X$4

At the same time, the example states that the degrees with respect to $X$5 vanish, so the numerical data fail to distinguish the two classes.

The role of this example is not peripheral. It shows that the passage from Chow-class equality to equality of intersection numbers is delicate, and that ampleness is part of the theorem’s content rather than a disposable hypothesis. A common simplification would be to replace ample by big and nef; the example shows that this is false in general.

5. Application to homogeneous ideals and Aluffi’s Segre zeta function

The criterion has a projective-space application to homogeneous ideals in polynomial rings. Let

$X$6

be homogeneous ideals. The paper applies the Segregativity Criterion to Aluffi’s Segre zeta function and proves that it provides an integral dependence criterion for homogeneous ideals (Cid-Ruiz, 9 Dec 2025).

The Segre zeta function is written as

$X$7

with the property that, for each $X$8,

$X$9

where $W \subseteq Z \subseteq X$0 is the closed subscheme of $W \subseteq Z \subseteq X$1 defined by extending $W \subseteq Z \subseteq X$2 to $W \subseteq Z \subseteq X$3, and $W \subseteq Z \subseteq X$4 is the class of a hyperplane in $W \subseteq Z \subseteq X$5.

The paper records that this zeta function is always a rational function: $W \subseteq Z \subseteq X$6 where $W \subseteq Z \subseteq X$7 are the degrees of the generators of $W \subseteq Z \subseteq X$8, and $W \subseteq Z \subseteq X$9 is a polynomial with nonnegative integer coefficients.

The resulting corollary is: $I_W \subseteq I_Z \subseteq \mathcal{O}_X .$0 Accordingly, the Segre zeta function is described as a complete invariant for integral closure of homogeneous ideals in a polynomial ring.

This application converts a global intersection-theoretic criterion into a generating-function criterion. It also shows that the Segre-class data can be organized across ambient projective dimensions without losing their sensitivity to integral closure.

6. Interpretation, significance, and scope

The paper’s summary identifies three principal consequences of the criterion (Cid-Ruiz, 9 Dec 2025). First, Segre classes serve as a global, intersection-theoretic measure of integral dependence. Second, the Segre zeta function encodes all Segre data across all dimensions, and equality of zeta functions is both necessary and sufficient for integral closure of homogeneous ideals in polynomial rings. Third, the criterion generalizes classical local results (like Rees’ criterion via Hilbert-Samuel multiplicity) to a global intersection-theoretic and projective setting.

Within this framework, the criterion can be understood as a bridge between commutative algebra and intersection theory. Integral closure is classically expressed in terms of algebraic conditions on ideals, valuations, or multiplicities. The Segregativity Criterion recasts the same relation in terms of Segre classes, Vogel cycles, polar cycles, and projective degrees.

The term itself is specific to this Segre-class context. The broader literature represented in the supplied corpus uses similar language for unrelated notions, including separability criteria in quantum information and segregation criteria in social-scientific models. This suggests that, in algebraic geometry, Segregativity Criterion names a precise theorem about Segre classes and integral dependence rather than a generic notion of separation or segregation.

In that sense, the criterion’s distinct contribution is conceptual as much as technical: it identifies the Segre class as exactly the right birationally invariant object to detect when one defining ideal sheaf is integral over another.