Segre classes of monomial schemes

We propose an explicit formula for the Segre classes of monomial subschemes of nonsingular varieties, such as schemes defined by monomial ideals in projective space. The Segre class is expressed as a formal integral on a region bounded by the corresponding Newton polyhedron. We prove this formula for monomial ideals in two variables and verify it for some families of examples in any number of variables.


Introduction
1.1. The excess numbers of a subscheme S of projective space P n are roughly defined as the numbers of points of intersection in the complement of S of n general hypersurfaces of given degrees containing S. Many challenging open enumerative problems, such as the problem of computing characteristic numbers for families of plane curves, may be stated in terms of excess numbers. Recently, the problem of computing excess numbers has been raised in algebraic statistics and in view of applications to machine learning and ideal regression.
The excess numbers of a subscheme S may be computed from the push-forward of the Segre class s(S, P n ) to P n . Segre classes are defined for arbitrary closed embeddings of schemes, and in a sense carry all the intersection-theoretic information associated with the embedding ( [Ful84], Chapters 4 and 6). Thus, they provide a general context applying in particular to the computation of excess numbers, and relating this problem directly with the well-developed tools of Fulton-MacPherson intersection theory. On the other hand, the computation of Segre classes is challenging, and indeed the connection with excess numbers appears to have mostly been exploited in the reverse direction-providing algorithms for the computation of Segre classes starting from the explicit solution of enumerative problems by computer algebra systems such as Macaulay2 ( [GS]). This strategy informs the author's implementation of an algorithm for Chern and Segre classes of subschemes of projective space in [Alu03], as well as more recent work on algorithmic computations of these classes ( [DREPS11], [EJP13]). As is usually the case with applications of computer algebra, such methods provide very useful tools for experimentations in small dimension, but do not lead to general results.
In this note we conjecture a general formula for the Segre class of a monomial subscheme, in terms of a corresponding Newton polyhedron. The monomial case is of independent interest, and in principle more general situations can be reduced to the monomial case by means of algebraic homotopies ( [Rod12]). We prove the formula in the case of monomials in two variables in any nonsingular variety, and verify it for some nontrivial examples in arbitrarily many variables. The formula is expressed as a formal integral over the region bounded by a Newton polyhedron associated with the subscheme. This integral can be computed directly from a subdivision of the region into simplices.
1.2. We now state the proposed formula precisely. Let V be a nonsingular variety, and let X 1 , . . . , X n be nonsingular hypersurfaces meeting with normal crossings in V . For I = (i 1 , . . . , i n ), we denote by X I the hypersurface obtained by taking X j with multiplicity i j , and call this hypersurface a 'monomial' (supported on X 1 , . . . , X n ). A monomial subscheme S of V is an intersection of monomials X I k supported on a fixed set of hypersurfaces. The exponents I k determine a (possibly unbounded) region N in the orthant R n ≥0 in R n , namely, the complement of the convex hull of the union of the orthants with origins translated at I k . We call this region the Newton region for the exponents I k .
Example 1.1. For n = 2 and monomials X (2,6) , X (3,4) , X (4,3) , X (5,2) , X (7,0) , the Newton region N is as in the following picture: N The third monomial X (4,3) does not affect the Newton region, as it is contained in the convex hull of the other translated quadrants. (Cf. Remark 2.5.) n!X 1 · · · X n da 1 · · · da n (1 + a 1 X 1 + · · · + a n X n ) n+1 The right-hand side is interpreted by evaluating the integral formally with X 1 , . . . , X n as parameters; the result is a rational function in X 1 , . . . , X n , with a well-defined expansion as a power series in these variables. The claim in Conjecture 1 is that evaluating the terms of this series as intersection products of the corresponding divisor classes in V gives the push-forward ι * s(S, V ).
Example 1.2. Using Fubini's theorem (!) to perform the integral for the monomials in Example 1.1 and taking X 1 = X 2 = H (for example, the hyperplane class in projective space) gives See Example 1.4 below for an alternative way to evaluate this integral. Expanding as a power series, According to Conjecture 1, the scheme S defined by the monomial ideal I = (x 2 1 x 6 2 , x 3 1 x 4 2 , x 4 1 x 3 2 , x 5 1 x 2 , x 7 1 ) in e.g., P 5 has Segre class ι * s(S, P 5 ) = provided that the hypersurfaces cut out S in a neighborhood of S. The excess number is the difference between this number and the Bézout number d 1 · · · d 5 .
For these examples we can compute independently the Segre class using the relation between Segre classes of singularity subschemes and Chern-Schwartz-MacPherson classes ([Alu99]), and we find (Proposition 3.4) that the expression we obtain does match the result of applying the formula given in Conjecture 1.
1.4. Excess numbers of monomial ideals admit an expression in terms of mixed volumes of polytopes, via Bernstein's theorem; the example of the monomial ideal (x p 1 1 , . . . , x p k k ) is worked out explicitly in [Rod12]. Thus, the expression obtained in (2) may be interpreted as a computation of the mixed volumes of certain polytopes in terms of the integral in (1), for the monomial subscheme of Example 1.1. Conversely, Bernstein's theorem may offer a path to the proof of the conjecture stated above for n > 2, at least if the classes X i are ample enough. We do not pursue this approach here; Bernstein's theorem is not used in our proof of Theorem 1.3.
A precise relation between Segre classes, volumes of convex bodies, and integrals such as those appearing in Conjecture 1 would be very valuable. Formula (1) (if verified) suggests that the Segre class of the scheme defined by an ideal I may be computed as a suitable integral over a region in R n ≥0 associated with I. The natural guess is that the convex bodies appearing in the work of Lazarsfeld and Mustaţȃ ( [LM09]) and Kaveh and Khovanskii ([KK12]) would play a key role in such a result.
As mentioned above, current algorithms for Segre classes essentially reduce the computation to enumerative problems, which are then solved by methods in computer algebra. This limits substantially the scope of these algorithms, and runs against one of the main applications of Segre classes: in principle one would want to compute Segre classes in order to solve hard enumerative problems, not the other way around. Formulas such as (1) do not rely on enumerative information, so they have the potential for broader applications.
1.5. We end this introduction by noting that the integrals appearing in (1) have the following property: if T is an n-dimensional simplex, then (3) T n!X 1 · · · X n da 1 · · · da n (1 + a 1 X 1 + · · · + a n X n ) n+1 = n! Vol(T ) X 1 · · · X n (a 1 ,...,an) (1 + a 1 X 1 + · · · a n X n ) where the product ranges over the vertices of T . An analogous expression may be given for unbounded regions dominating a simplex in lower dimension; see Proposition 3.1. Thus, one may compute the integral in (1) by splitting the region N in any way into (possibly unbounded) simplices and applying (3).
Example 1.4. The computation carried out in Example 1.2 may be performed by splitting the region N as a union of triangles and one unbounded region as follows: Each triangle contribues a rational function according to (3): (the first term accounts for the unbounded region). This reproduces the result obtained in Example 1.2.
This approach may in fact be taken as an alternative interpretation of the meaning of the right-hand side of (1). What is possibly surprising from this point of view, and is transparent from the interpretation as an integral, is that the result does not depend on the chosen decomposition.
Theorem 1.3 is proven in §2. Generalizations of (3) and examples giving evidence for the validity of (1) for monomial ideals in arbitrarily many variables are discussed in §3.
1.6. Acknowledgments. The author's research is partially supported by a Simons collaboration grant.
(Again, this is a simple induction; see Proposition 3.1 for generalizations.) Now consider the subscheme S ι → V defined by a principal ideal (X I ) generated by a single monomial, with I = (i 1 , . . . , i n ). Then S is a Cartier divisor, with normal bundle and therefore The integral from equation 1 for this example is n! X 1 · · · X n da 1 · · · da n (1 + a 1 X 1 + · · · + a n X n ) n+1 = 1− 1 1 + i 1 X 1 + · · · + i n X n = i 1 X 1 + · · · + i n X n 1 + i 1 X 1 + · · · + i n X n verifying Conjecture 1 for principal monomial ideals.
In particular, this verifies Conjecture 1 for n = 1.
2.2. n = 2. Monomial ideals in two variables can be principalized by a sequence of blow ups along codimension 2 loci. (In fact, every monomial ideal may be principalized by blowing up codimension 2 loci, cf. [Gow05].) A principalization algorithm may be described as follows: if S is a monomial ideal supported on X 1 , X 2 in V , let π : V → V be the blow-up of V along X 1 ∩ X 2 (which is nonsingular by hypothesis), let E be the exceptional divisor, and X 1 , X 2 the proper transforms of X 1 , X 2 . (Note that X 1 , X 2 are disjoint.) Up to a principal component supported on E, π −1 (S) is again a union of disjoint monomial subschemes S 1 , S 2 . Iterating this process on S 1 , S 2 leads to a sequence of blow-ups principalizing S. Therefore, the n = 2 case of Conjecture 1 may be proven by showing that the validity of the conjecture for S 1 , S 2 implies that (1) holds for S: indeed, induction on the number of blow-ups needed for a principalization reduces the conjecture to the principal case, which has been verified in §2.1. We carry out this strategy in the rest of this section. Let S be defined by monomials X I k , with I k = (i k1 , i k2 ), k = 1, . . . , r. The Newton region N is the complement of the convex hull of the union of the (i k1 , i k2 ) translations of the positive quadrants: where ι : S → V is the embedding. With π : V → V as above, let j be the inclusion of π −1 (S) in V . The residual of mE in π −1 (S) consists of two disjoint subschemes S 1 , S 2 of V . These are monomial subschemes, supported respectively on X 1 , E and E, X 2 .
Proof. In a neighborhood of S 1 , π −1 (S) is the intersection of divisors denoted additively as i k1 π −1 (X 1 ) + i k2 π −1 (X 2 ) = i k1 ( X 1 + E) + i k2 E = i k1 ( X 1 ) + (i k 1 + i k 2 )E (since S 1 is disjoint from X 2 , π −1 (X 2 ) agrees with E near S 1 ). The monomial scheme S 1 is obtained as the residual of mE in this intersection, with corresponding exponents as stated. The analysis is identical near S 2 .
(Here and in what follows we use freely notation as in [Alu94], §2.) Therefore, i * s(S, V ) is naturally the sum of three terms: the push-forwards to V of The following claim will conclude the proof of Theorem 1.3.
Claim 2.2. The terms (i), resp. (ii), (iii) push-forward to the values of the integral on the subregions T , resp. N , N of N determined above.
Proof. -(i): By the birational invariance of Segre classes, the push-forward of E/(1 + E) is the Segre class of the center of the blow-up. Therefore, the push-forward of mE/(1 + mE) is the m-th Adams of this Segre class. Since the center is the complete intersection of X 1 and X 2 , this is given by .
(The Segre class of a complete intersection equals the inverse Chern class of its normal bundle.) The claim is that this expression equals the integral T 2X 1 X 2 da 1 da 2 (1 + a 1 X 1 + a 2 X 2 ) 3 where T is the triangle with vertices (0, 0), (0, m), (m, 0). The verification of this fact is a trivial calculus exercise. (See Proposition 3.1 for a generalization.) -(ii): Term (ii) is where S 1 is monomial on X 1 , E with exponents (i k1 , i k1 + i k2 − m) as observed in Claim 2.1. Each vertex of the Newton polyhedron for S determines a vertex for S 1 by the transformation (a 1 , a 2 ) → (ã, e) = (a 1 , a 1 + a 2 − m). (1 +ã X 1 + eE) 3 where ι 1 : S 1 → V is the embedding, and N 1 is the Newton region for S 1 . Note that N 1 maps onto region N via the transformation (ã, e) → (a 1 , a 2 ) = (ã, e −ã + m).
Proof. Applying (4), the projection formula, and the fact that the line bundle O(mE) is constant with respect to the integration variablesã and e, we see that the left-hand side equals (1 +ã X 1 + (e + m)E) 3 as stated. (We have used here the formal properties of the ⊗ operation, in particular Proposition 1 of [Alu94].) Thus, we are reduced to proving Claim 2.4.
-(iii) is handled in exactly the same way.
Remark 2.5. As a consequence of Theorem 1.3, monomial generators which do not affect the Newton region (i.e., which are in the convex hull of the translated quadrants determined by the other generators) do not affect the Segre class.
This fact is not surprising, and holds for arbitrary n. Indeed, such generators do not affect the integral closure of the ideal, and the Segre class only depends on the integral closure, cf. the proof of Lemma 1.2 in [Alu95].
Let T be a k-dimensional simplex in R n , with vertices v 0 , . . . , v k . For J = {j 1 , . . . , j k }, with 1 ≤ j i < · · · < j k ≤ n, let π J : R n → R k be the projection to the span of e j 1 , . . . , e j k . We denote by T J the region {a + i ∈J λ i e i | a ∈ T, λ i ≥ 0}.
Performing the unbound integrations shows that this equals Thus, the statement of Proposition 3.1 is reduced to the following fact: with M, L 1 , . . . , L k independent of t. This equality is immediately verified by induction.
With X 1 = X 2 = H, this gives the term 2H 1 + 8H used in Example 1.4.
This will be used below in Example 3.3.
This verifies Conjecture 1 for these complete intersection.
A somewhat harder calculus exercise verifies Conjecture 1 for arbitrary complete intersections of monomials. As an example of what is involved in this verification, consider a monomial subscheme S ι → V supported on X 1 , . . . , X n−1 , and let S be the intersection of S with the m-multiple mX n . Standard facts about Segre classes imply that (7) ι * s(S , V ) = mX n 1 + mX n ∩ ι * s(S, V ) .
To see that Conjecture 1 is compatible with this formula, observe that the Newton region N for S is a cone over the Newton region N for S with vertex at (0, · · · , 0, m).

N N'
The equality (7) amounts to N n! X 1 · · · X n da 1 · · · da n (1 + a 1 X 1 + · · · + a n X n ) n+1 = mX n 1 + mX n ∩ N (n − 1)! X 1 · · · X n−1 da 1 · · · da n−1 (1 + a 1 X 1 + · · · + a n−1 X n−1 ) n and in the cone situation this follows from with suitable positions for A and B, as the reader may check. A similar (but harder) computation verifies the corresponding formula whenever S is a monomial scheme in X 1 , . . . , X k and the lone vertex is a single monomial in X k+1 , . . . , X n . Repeated application of this more general formula implies that (1) holds for any complete intersection of monomials.
3.3. Singularity subschemes. As a less straightforward family of examples verifying Conjecture 1, we consider the monomial subschemes on X 1 , . . . , X n with exponents These subschemes are very far from being complete intersections (for n > 1), and computing their Segre class requires some nontrivial work, which depends on features of these schemes which do not hold for arbitrary monomial schemes. Ad-hoc alternative methods are occasionally available, as in the following example.
(1,0,1) (0,1,1) The contribution of a column to the integral in (1) was computed in Example 3.2, and equals (setting X 1 = X 2 = H, the hyperplane class) The tetrahedron has volume 1 3 , hence it contributes (using Proposition 3.1, and again setting Therefore, according to Conjecture 1, the Segre class equals For any n, the subscheme S defined by the exponents (8) is the singularity subscheme of the union X of the hypersurfaces X 1 , . . . , X n , i.e., the subscheme of X locally defined by the partials of an equation for X in V . (For example, (xy, xz, yz) is the ideal generated by the partials of xyz.) This is what gives us independent access to the Segre classes for these subschemes, and allows to verify Conjecture 1 in these cases.
Proof. Let F = {f 1 , . . . , f n } be the set of exponents. The Newton region N may be described as follows. For any J ⊆ {1, . . . , n}, let Σ J be the simplex with vertices at the origin and at the points f j with j ∈ J, and consider the subsets Σ J J , with notation as in §3.1. The reader can verify that the Newton region N is then the union of the sets Σ J J with |J| ≥ 2. (For example, the region N for n = 3 decomposes as the union of the three columns Σ 12 12 , Σ 13 13 , Σ 23 23 , and the 3-simplex Σ 123 123 . Cf. Example 3.3.) The volume of the projection π J (Σ J ) is easily found to be (|J| − 1)/|J|!. By Proposition 3.1, N n! X 1 · · · X n da 1 · · · da n (1 + a 1 X 1 + · · · + a n X n ) n+1 = |J|≥2 (|J| − 1) j∈J X j j∈J (1 + f j · X) .
We have to verify that this equals the class ι * s(S, V ). This Segre class is computed in [Alu99] where L i = O(X i ), L = O(X 1 + · · · + X n ). (This is an instance of the relation between the Chern-Schwartz-MacPherson class of a hypersurface and the Segre class of its singularity subscheme, in the particular case of divisors with normal crossings.) Thus, we have to verify that (9)