Effective Intersection Theory

06/19/2018
by   Corey Harris, et al.
UNIVERSITETET I OSLO
0

Let X ⊂ Y be closed (possibly singular) subschemes of a smooth projective toric variety T. We show how to compute the Segre class s(X,Y) as a class in the Chow group of T. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of T. Our methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.

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1. Introduction

Segre classes capture important enumerative and geometric properties of systems of polynomial equations coming from embeddings of schemes. Historically, these classes have played a fundamental role in the development of Fulton-MacPherson intersection theory [Ful98, §6.1]. Computation of Segre classes (other than in a few special cases) has proven to be a challenge; this has limited the development of applications in practice.

Evidence of the significance of Segre classes in algebraic geometry can be found in the fact that many important characteristic classes can be written as , where

is some polynomial in the Chern classes of vector bundles on

. The flagship example of this is the topological Euler characteristic , which appears in the Chern-Schwartz-MacPherson class . By results of Aluffi [Alu03, Alu18] the class can be directly obtained by computing a Segre class. There are also formulas in terms of Segre classes for the Milnor class of a hypersurface [Alu03], the Chern-Mather class and polar degrees [Pie78], and the Euclidean distance degree of a projective variety [AH17].

More generally, many enumerative problems end up in the situation of an excess intersection, in which an intersection is expected to be finite but instead is the union of a finite set of points along with a positive-dimensional set. Typically the desired quantity is then the number of points outside the positive-dimensional part. The Segre class gives a way to express the contribution of this part, which is the difference between the expected number (e.g., the Bézout bound) and the actual number of points in the finite set.

Let be closed subschemes of a smooth projective toric variety . The Segre class of in is a class in the Chow group of . Since the group is often unknown, the best we could hope for in general is to compute the pushforward of this class to .

Previous work on computing Segre classes of the form in (i.e., the special case where ) began with the paper [Alu03] and alternative methods were developed in [EJP13, Hel16]. These methods were generalized to compute in in [MQ13, Hel17]. In [Har17] the scope was extended to compute the Segre class pushed forward to for and a variety. The present work goes further by taking arbitrary subschemes not just of projective space, but of any smooth projective toric variety, to obtain in .

The ability to effectively compute Segre classes in this new setting opens the way for several novel computational applications. For instance, computing Segre classes in products allows for a general framework to compute intersection products in subvarieties of . While a big part of this paper is devoted to studying Segre classes, many of the resulting applications can be expressed without them. In particular, we show that algebraic multiplicity can be computed and pairwise containment of varieties can be tested by counting the number of points in a single zero-dimensional set.

1.1. Examples

We now give three examples which illustrate how the results developed in this paper give rise to new methods to answer classical geometric and enumerative questions. The first example shows how the intersection product can be computed using a Segre class. Following this we give examples which compute algebraic multiplicity and test ideal containment. The methods presented in the later two examples build on ideas developed to compute Segre classes (which we present in §3), but can be understood without them.

1.1.1. Intersection theory

The intersection product of varieties and in a non-singular variety , denoted (see §4 for a definition), captures the behavior of the intersection inside of . It is a class in the Chow ring . If meets transversely in the expected dimension, the intersection product may be defined by

If and do not meet dimensionally transversely, but there exist and which are respectively equivalent in and are dimensionally transverse, then we define the intersection analogously:

Figure 1. Lines on a quadric.
Example 1.1 (Lines on a quadric, I).

The quadric surface in Figure 1, comes with two families of lines, and , and any two lines in the same family are equal in . Let and be two such lines on . We compute and .

Since meets only at and the intersection is transverse of the expected dimension , we can conclude that is the class of a point.

In contrast, does not have the expected dimension, so we can try to “move” one of the terms by finding a suitable replacement with the same class in . If is any other line, then is empty, and thus the intersection product is 0.

Another way to perform the computation above would be to recognize that and , . Then the product in this ring is the intersection product, i.e.  and . However, when computing in practice, one may encounter a variety for which the Chow ring is not known a priori. In this situation, using the methods of §3-4, we could still arrive at the answer.

Example 1.2 (Lines on a Quadric, II).

Let and forget that is a smooth quadric. Given subvarieties and defined on by and , respectively, we wish to compute the intersection products and . Since we do not know the Chow ring, we must do something new. Specifically we work in the Chow ring of the ambient , which is , and write the intersection product in terms of classes there.

In §4, we will see that a universal setup for this approach is to intersect with the diagonal in . Then the ingredients to our computation are:

  1. the Chern class , where is the inclusion,

  2. the Segre class ,

  3. the Segre class ,

where is the diagonal map and

are the hyperplane classes for each factor. As above, we will frequently suppress obvious pushforwards. Then, as follows from Theorem

4.1, we confirm and .

Remark 1.3.

In Example 1.2 we did not need to do any computations by hand. Items (i)-(iii) can all be computed in a computer algebra system given the defining equations. Following [Alu03], is determined also by a Segre class (specifically the Segre class ), which could already be found via the methods of [Alu03, EJP13, Hel16], and computing items (ii) and (iii) are contributions of this paper.

1.1.2. Algebraic multiplicity

Our second example illustrates how Segre classes can be used to compute the algebraic multiplicity of a local ring with respect to an ideal (without computing in the local ring).

Example 1.4 (Algebraic multiplicity along a component).

Let be the homogeneous coordinate ring of . As in [Say17, Ex. III.10], we consider the twisted cubic defined by the prime ideal

and the scheme defined by the ideal

Then is a subscheme of , since . Let be the local ring of along . The algebraic multiplicity of along is the leading coefficient of the Hilbert-Samuel polynomial associated to the local ring (see §5). The multiplicity may be read off of the Segre class since is also the coefficient of in the class , see Definition 5.1. The class is . Applying the methods of §3 we find that

Together this gives that . Let be the maximum degree among the defining equations of the ideals and and let be the dimension- projective degree defined by in (see §2.3). More directly, by Theorem 5.3 we have that

1.1.3. Containment

Our third example demonstrates a new criterion to test containment of varieties or of irreducible components of schemes (see §6).

Example 1.5 (Containment of Varieties).

Work in with coordinates and let

The variety is an irreducible singular surface of degree 20 in .

Now set and . We seek to determine if the line is contained in the singular locus of the surface . The standard method to test for this containment is to compute the ideal defining and reduce each generator of with respect to a Gröbner basis for . In this case the ideal is defined by the minors of the Jacobian matrix of ; it is clear from the structure of (it has generators of degree three in six variables) that the computation of the minors to obtain the ideal will be very time consuming.

On the other hand, by Corollary 6.2 we have that if and only if

where the left-hand side is (computed via Theorem 5.3), and is the dimension- projective degree of in (see §2.3). Using Theorem 3.5 we compute by finding the number of solutions to a single zero-dimensional system of polynomials, with each polynomial of degree at most three. Substituting this in we have that and, hence, . The computation of the integer takes approximately seconds using Macaulay2 [GS] on a laptop.

We note again that the test described above using Corollary 6.2 does not compute the ideal defining . In this case computing the ideal and using Gröbner basis methods to test if takes approximately 692 seconds using Macaulay2 [GS] on the same test machine (the majority of this time, about 690 seconds, is spent computing the ideal ).

This paper is organized as follows. In Section 2 we establish our notation and conventions and present relevant background on Segre classes and projective degrees. The main results of this paper are presented in Section 3. In §3.1 we consider the case where is a subscheme of an irreducible scheme . In Theorem 3.6 we give an explicit formula for the Segre class in terms of the projective degrees of a rational map defined by from to a projective space (see §2.3). In Theorem 3.5 we give an expression for these projective degrees as a vector space dimension of a ring modulo a certain zero-dimensional ideal. These results are generalized in §3.2 to the case where is a subscheme of a smooth projective toric variety .

In Section 4 we show how the results of Section 3 can be applied to compute intersection products. If is a smooth variety, requires computing . The main result is that the pushforward of this class to is enough to recover the pushforward of the intersection product.

In Section 5 we use the results of §3.1 to give an explicit expression for , the algebraic multiplicity of along , in terms of ideals in the coordinate ring of . The expressions are also generalized to .

Finally in Section 6 we combine the results from §5 with a classical result of Samuel [Sam55] to yield new numerical tests for the containment of one variety in another. Let and be arbitrary subvarieties of a smooth projective toric variety . In §6.1 we give a simple criterion to determine if is contained in the singular locus of without computing the defining equations of the singular locus. In §6.2 we give a criterion to determine if . As in the previous results, computing a Gröbner basis is not required and methods from numerical algebraic geometry could be used. To the best of our knowledge, this is the first general purpose method which is able to test containment of possibly singular varieties using only numeric methods.

As of version 1.13, Macaulay2 [GS] contains the SegreClasses package, which implements many of the results described in this paper.

2. Background

In this section we review several definitions and explicitly state the notations and conventions we will use throughout the paper.

2.1. Notations and conventions

We work throughout over an algebraically closed field of characteristic zero.

2.1.1. Varieties, schemes, and irreducible components

Since we always work in an ambient projective variety, all schemes will be of finite type over the base field. By variety we mean a reduced and irreducible, separated scheme, that is, an irreducible algebraic set. Given polynomials we let denote the algebraic set defined by . Conversely, if is a subscheme of a smooth variety with coordinate ring , we will let be the ideal in defining the scheme and let be the radical ideal in defining the reduced scheme . If is a subscheme of a smooth variety defined by an ideal , its primary components are the schemes associated to the primary components of ; its irreducible components are the varieties defined by the associated primes of .

2.1.2. Chow classes

Let be a subscheme of a smooth variety . The irreducible components of have associated geometric multiplicity given by the length of the local ring , and we write for the rational equivalence class of in . We frequently write to mean the pushforward via inclusion. For a cycle class in the Chow group we will use the notation to denote the degree of the zero-dimensional part of (as in Definition 1.4 of Fulton [Ful98]). The degree of a zero-dimensional scheme is . For instance, if is defined by then .

2.1.3. The total coordinate ring of a toric variety

Let be a smooth projective toric variety defined by a fan and let denote the rays in the fan. The Cox ring of is . The ring can be graded by defining the multidegree of a monomial to be , where denotes the codimension-one Chow group of . Setting we have that We say that a polynomial is homogeneous if it is homogeneous with respect to this grading, i.e., if all monomials in have the same multidegree. An ideal in is called homogeneous if it is generated by homogeneous polynomials. When the Cox ring is simply the standard graded coordinate ring of and the multidegree of a monomial is simply the total degree of the monomial multiplied by the class of a general hyperplane. More details can be found in the book [CLS11].

2.1.4. Homogeneous generators

Let be a closed subscheme. We can always find an ideal defining so that for all , for some fixed in (see [CLS11, 6.A]). Using this set of generators we see that is the base scheme of the linear system defined by , viewed as sections of .

Definition 2.1.

We say a homogeneous polynomial in the Cox ring of a smooth projective toric variety has multidegree and say a set of polynomials all having the same multidegree is -homogeneous.

Convention 2.2.

Let be a subscheme of defined by an ideal generated by polynomials , we assume (without loss of generality) that this set of polynomials is -homogeneous. We will use this convention for the defining equations of all subschemes/subvarieties considered in this paper unless otherwise stated.

Remark 2.3.

In the case where we are simply assuming that a given set of polynomial generators have the same degree. If we are given an ideal in the coordinate ring of where the generators do not have the same multidegree we may construct a new ideal which has -homogeneous generators and defines the same scheme as follows.

Work in with multi-graded coordinate ring , where . In this case the Chow group is generated by where is the pullback of the hyperplane class in the factor . The multidegree of a monomial in has the form where is the total degree of in the variables . Let , so that the irrelevant ideal of has primary decomposition , and for , let be the ideal generated by the -th powers of the generators of .

Take a subscheme defined by a homogeneous ideal in with the generator having multidegree . Let for . We can construct a new ideal for with generators all having multidegree ; the ideal is given by:

Note that in the equation above we use summation notation for the sum of ideals in . The reader can verify that , meaning that and define the same subscheme.

Example 2.4.

The procedure discussed in Remark 2.3 above is easy to apply by hand. Work in with coordinate ring and codimension-one Chow group generated by the hyperplane classes . The irrelevant ideal of the coordinate ring is

Consider the ideal defining a scheme . The generators of have multidegrees and , respectively. The new ideal will have generators all of multidegree :

and the ideal also defines the scheme since .

To construct an -homogeneous set of generators from a given set of homogeneous generators for a subscheme of an arbitrary smooth projective toric variety, a technique similar to that of Remark 2.3 can be used. Instead of multiplying by powers of the generators of components of the irrelevant ideal, we multiply the given generators by powers of the generators of products of components of the irrelevant ideal of the Cox ring. Since the fan of a general smooth projective toric variety is more combinatorially complicated than that of a product of projective spaces, the procedure is also more difficult to write down, but is otherwise similar.

2.1.5. Multi-indices

We will make frequent use of standard multi-index notations throughout the paper. In particular for a non-negative integer vector , we have , and if are the variables of a ring, we write . Notice that if are generators of the Chow ring , then is a class of codimension .

2.2. Segre classes

In this subsection we define the Segre class of a subscheme and summarize computational methods for Segre classes in the Chow ring .

For a subscheme of a scheme the Segre class is the Segre class of the normal cone to in (see [Ful98, §4.2] for more details). In the case where is a variety we may define the Segre class via Corollary 4.2.2 of [Ful98].

Definition 2.5.

Let be a closed subscheme of a variety . We have a blowup diagram

where is the exceptional divisor. The Segre class of in is

see [Ful98, Corollary 4.2.2]. When is contained in some smooth projective toric variety we will frequently abuse notation and write for the pushforward to .

We now review the computation of the class in the special case where is a subscheme of a projective variety . In this case we may think of as the base scheme of an -dimensional linear system of global sections of . In practice this means choosing a set of (scheme-theoretic) generators for all of the same degree , say . Consider the graph

(1)

of the rational map defined by .

Definition 2.6.

We call the projection of along . The projective degrees of are the non-negative integers

where is the hyperplane class in the Chow ring .

Proposition 2.7 ([Har17, Prop. 5]).

The Segre class is given by

In [Hel16, Theorem 4.1], the projective degrees of a rational map are expressed as the dimensions of a sequence of finite-dimensional -algebras. There is an analogous result for our more general situation. The projective degrees for can be computed directly from the definition above as the degree of the 0-dimensional variety

where is a general linear space of codimension in and is a general -linear combination of -homogeneous generators of . Note by construction.

We now move to and restrict to the affine open subset defined by the non-vanishing of a general linear form . Further, we want to remove , so we restrict to where is a general -linear combination of -homogeneous generators for . The resulting affine variety

is in 1-1 correspondence with the set of points . This leads to the following expression for the projective degrees.

Proposition 2.8 (cf. [Hel16, Theorem 4.1]).

The projective degrees of are given by

where is a general affine linear form in the and is a general -linear combination of -homogeneous generators for .

2.3. Projective degrees of in

Let be a smooth projective toric variety defined by a fan with Cox ring (see §2.1.3). Let be a pure-dimensional subscheme and let be a subscheme defined by an -homogeneous ideal (see Definition 2.1).

Definition 2.9.

We define the projection of along to be the rational map given by

(2)

Let denote the blowup of along . We have the following diagram:

(3)

Now we define the class

(4)

where is the pullback along of the hyperplane class. This class is a minor generalization of Aluffi’s shadow of the graph ([Alu03]). By construction

where is a general linear subspace.

We may write more explicitly as follows. Let be a fixed nef basis for (this exists since is projective, see [CLS11, Proposition 6.3.24]). We may express the rational equivalence class of a point in as where for all , and the degree- monomials in which divide in form a monomial basis for . Hence we may write

(5)

where . Note that the indices appearing in this expression depend on the choice of representative of the point class, which is not unique in general; however the class does not depend on this choice.

Definition 2.10.

We refer to the coefficients as the projective degrees of in . In other words the projective degrees are

(6)

The class measures, in a sense, how algebraically dependent the polynomials are in . If the linear system were base-point free on (so and ), the shadow of the graph would be . The difference of these two classes will play an important role in the computation of the Segre class (see Theorems 3.6 and 3.13), so we define the following notation:

(7)

In particular, when .

Example 2.11 (Projective degrees in ).

Work in with Chow ring and consider the varieties and . Using Theorem 3.5 we compute that the projective degrees are and . Hence we have that . For this example and we let . Substituting these values into (7) we obtain

Example 2.12 (Projective degrees in products of projective spaces).

Work in with Chow ring . In this Chow ring the class of a point is . Consider the 3-dimensional variety

with divisor defined by . For this example, , and and we let . Computing the projective degrees using Theorem 3.5 we obtain the following. In dimension zero we have . In dimension one we have and . In dimension two we have , , and . In dimension three we have , , and . Hence we have that

Substituting these values into (7) we obtain

Example 2.13 (Projective degrees in a toric variety).

Work in the smooth Fano toric threefold111The variety is generated in Macaulay2 [GS] with the command smoothFanoToricVariety(3,2) from the NormalToricVarieties package. where is the fan with rays , , , , and maximal cones , , , , , . Let denote the orbit closure of a cone and set and . The divisors form a nef basis for . The Cox ring of is , with irrelevant ideal . The Cox ring is graded; the multidegree of is , the multidegree of , and is , and the multidegree of is . The Chow ring can be written as

For more on constructing the Chow ring of a smooth projecive toric variety from its fan see [Dan78, Theorem 10.8].

Let and let . Then is a curve on the surface . For this example, and we let . Let denote the rational equivalence class of a point in . Using Proposition 3.11 we compute the projective degrees and obtain Substituting and into (7) we obtain

Note that . The lack of a unique representative for the point class in this case stems from the existence of non-effective divisors on . Since the indexing convention for projective degrees depends on the chosen representative of the point class, we have opted to simply write the resulting unique class .

Remark 2.14.

If is the complete intersection of general divisors of with such that , then we have that

In particular, if is a divisor with , then