Blow-Up Constructions and Applications to Segre Classes and Multidegree Formulas

Abstract: By comparing simultaneous multigraded blow-ups with suitable iterated blow-up construction, we establish a birational correspondence between the associated exceptional divisors. We further investigate blow-ups arising from rational maps to multiprojective spaces and derive intersection-theoretic formulas for the Chern classes of the resulting exceptional divisors and pullback of tautological line bundles. These constructions have two main applications. First, they yield a proof of the product formula for Segre classes without any pure-dimensionality hypothesis. Second, they lead to a degree formula for arbitrary closed subschemes of multiprojective spaces, recovering and extending the classical formula of van der Waerden.

• The paper establishes birational comparisons between simultaneous and iterated blow-ups, transferring exceptional-divisor cycles and enabling intersection-theoretic calculations.
• The paper proves the product formula for Segre classes for arbitrary closed embeddings, removing pure-dimensionality, integrality, and equidimensionality assumptions.
• The paper derives a multidegree-to-degree formula for arbitrary closed subschemes of multiprojective spaces, extending van der Waerden’s classical result through blow-ups and Chow theory.

Overview

The paper establishes two birational comparison results for blow-up constructions and applies them to obtain intersection-theoretic formulas. The first result compares the simultaneous multigraded blow-up of a product of closed embeddings with an iterated blow-up construction, producing a birational correspondence that pushes forward fundamental cycles of exceptional divisors. The second identifies the blow-up along a product of ideals with the scheme-theoretic closure of a rational map to a multiprojective space, yielding explicit relations among Chern classes of tautological line bundles and exceptional divisors. These tools are then applied to prove the product formula for Segre classes without any pure-dimensionality hypothesis, and to derive a degree formula for arbitrary closed subschemes of multiprojective spaces, extending the classical formula of van der Waerden [vdW78]. Throughout, the author works within Fulton's formalism of Chow groups, normal cones, and refined intersection products [Ful98].

Comparison of simultaneous and iterated blow-ups

Fix closed embeddings $X_i \hookrightarrow Y_i$ defined by coherent ideal sheaves $\mathcal{I}_i$, and let $Bl_i \to Y_i \times \mathbb{A}^1_k$ be the blow-up along $X_i$ embedded via the zero section, with exceptional divisor $E_i$. The paper considers two models: $Bl$, the blow-up of $\prod_i (Y_i \times \mathbb{A}^1_k)$ along $\prod_i X_i$, and $Bl'$, the blow-up of the regular embedding $$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$. The main structural result (Proposition 2.1) constructs morphisms $\mathcal{I}_i$0 and $\mathcal{I}_i$1 commuting with the blow-down maps such that:

• $\mathcal{I}_i$2;
• $\mathcal{I}_i$3;
• $\mathcal{I}_i$4.

The proof proceeds by the universal property of blow-ups to obtain $\mathcal{I}_i$5, and by analyzing generic points to establish equality of cycle classes. Since each center contains no generic points of its ambient space, both blow-ups are isomorphisms over a common dense open subscheme containing all generic points, which gives $\mathcal{I}_i$6. For the exceptional divisors, the argument is more delicate: $\mathcal{I}_i$7 is a $\mathcal{I}_i$8-bundle over $\mathcal{I}_i$9 (the center being a regular embedding), while $Bl_i \to Y_i \times \mathbb{A}^1_k$0 is described as the projectivization of a tensor product of the cones $Bl_i \to Y_i \times \mathbb{A}^1_k$1. On the dense open where all cone coordinates are nonzero, the induced morphism on Proj is explicitly given by a graded ring homomorphism sending $Bl_i \to Y_i \times \mathbb{A}^1_k$2 to $Bl_i \to Y_i \times \mathbb{A}^1_k$3, which is an isomorphism over $Bl_i \to Y_i \times \mathbb{A}^1_k$4; this yields the bijection on generic points with isomorphic local rings, hence $Bl_i \to Y_i \times \mathbb{A}^1_k$5.

This birational correspondence is the geometric mechanism underlying the first application: it transfers intersection-theoretic data from the iterated model, where Segre classes behave multiplicatively under products, to the simultaneous model.

Product formula for Segre classes without pure-dimensionality assumptions

Recall that for a closed embedding $Bl_i \to Y_i \times \mathbb{A}^1_k$6, the Segre class is defined via the projectivization of the normal cone:

$Bl_i \to Y_i \times \mathbb{A}^1_k$7

Existing proofs of the product formula generally require the schemes involved to be pure-dimensional, because standard cycle-theoretic relations in [Ful98] are invoked under that hypothesis. The paper removes this assumption entirely.

The key observation is that $Bl_i \to Y_i \times \mathbb{A}^1_k$8 for any closed embedding, proved directly from the definition using Lemma 1.7.2 of [Ful98]: the extra hyperplane summand contributes nothing since $Bl_i \to Y_i \times \mathbb{A}^1_k$9 for dimensional reasons. With this stabilization in place, the proof of multiplicativity proceeds as follows. For regular embeddings, the formula follows immediately from the splitting of the normal bundle into the direct sum of pullbacks of the factors' normal bundles. For general embeddings, one applies Proposition 2.1: the Segre class of the product embedding equals the pushforward from the exceptional divisor $X_i$0 of the simultaneous blow-up, which by $X_i$1 and the projection formula equals the pushforward from $X_i$2; but $X_i$3, and since each $X_i$4 is a regular embedding, the already-established regular case gives

$X_i$5

The conceptual point is that the exceptional divisor serves as a bridge reducing arbitrary closed embeddings to Cartier divisors on a suitable resolution, so the product formula becomes a consequence of the geometry of blow-ups rather than of cycle-theoretic identities requiring dimension hypotheses. This is a genuine strengthening: no integrality or equidimensionality of the $X_i$6 or $X_i$7 is assumed.

Graph closures versus blow-ups

The second part of the paper studies a rational map defined by global sections $X_i$8, not all identically zero. These determine an ideal sheaf $X_i$9 (the image of the evaluation map twisted by $E_i$0), a rational map $E_i$1 on $E_i$2, and two compactifications: the blow-up $E_i$3 and the graph closure $E_i$4, the scheme-theoretic image of $E_i$5. A surjection of graded algebras $E_i$6 produces a closed embedding of $E_i$7 into $E_i$8 with $E_i$9, and a natural closed embedding $Bl$0.

When $Bl$1 is integral, the embedding $Bl$2 is an isomorphism, shown via injectivity of $Bl$3 in a short exact sequence diagram. The multi-section version (Proposition 3.1) extends this: for integral $Bl$4 with invertible sheaves $Bl$5 and sections defining ideals $Bl$6, the blow-up along the product ideal $Bl$7 is identified with the graph closure of $Bl$8, and there exist morphisms $Bl$9 satisfying

$\prod_i (Y_i \times \mathbb{A}^1_k)$0

and consequently $\prod_i (Y_i \times \mathbb{A}^1_k)$1.

The identification rests on the fiber-product decomposition $\prod_i (Y_i \times \mathbb{A}^1_k)$2 together with the surjection onto $\prod_i (Y_i \times \mathbb{A}^1_k)$3. Note that integrality of $\prod_i (Y_i \times \mathbb{A}^1_k)$4 is used here; the paper does not claim the graph-closure identification for non-reduced or reducible bases.

Multidegree formulas and the extension of van der Waerden's theorem

Specializing to $\prod_i (Y_i \times \mathbb{A}^1_k)$5 with two sets of homogeneous coordinate sections, the construction yields a blow-up $\prod_i (Y_i \times \mathbb{A}^1_k)$6 embedded in $\prod_i (Y_i \times \mathbb{A}^1_k)$7. Because every fiber of $\prod_i (Y_i \times \mathbb{A}^1_k)$8 is isomorphic to $\prod_i (Y_i \times \mathbb{A}^1_k)$9, Miracle Flatness implies this map is smooth proper of relative dimension one. Writing $\prod_i X_i$0 and $\prod_i X_i$1 for the pullbacks of the hyperplane classes, the disjointness of $\prod_i X_i$2 and $\prod_i X_i$3 forces

$\prod_i X_i$4

from which powers of $\prod_i X_i$5 reduce via complete homogeneous symmetric polynomials:

$\prod_i X_i$6

with $\prod_i X_i$7 for $\prod_i X_i$8 and $\prod_i X_i$9.

For a closed subscheme $Bl'$0, let $Bl'$1 and let $Bl'$2 be the scheme-theoretic image of the inverse image on the open locus. Lemma 1.7.1 of [Ful98] gives $Bl'$3 in every dimension, compatibly with cycle decompositions. The central result (Proposition 3.2) states that for every $Bl'$4, the codimension-$Bl'$5 degree of $Bl'$6 equals the sum of the codimension-$Bl'$7 bidegrees of $Bl'$8 — for arbitrary closed subschemes, not merely varieties. The proof pushes forward $Bl'$9 through $$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$0 using the projection formula and the quadratic relation; the term involving $$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$1 computes exactly the sum of bidegrees (all fibers being $$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$2), while the $$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$3 term vanishes for dimensional reasons. The paper notes the evident generalization to arbitrary multiprojective spaces: codimension-$$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$4 degrees equal sums of codimension-$$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$5 multidegrees.

This recovers van der Waerden's classical degree formula [vdW78]. When $$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$6 is integral of dimension $$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$7, $$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$8 is integral of dimension $$E_1 \times \cdots \times E_n \hookrightarrow Bl_1 \times \cdots \times Bl_n$$9, and the bihomogeneous equations of $\mathcal{I}_i$00 define $\mathcal{I}_i$01 in $\mathcal{I}_i$02 — precisely van der Waerden's setting. The modern derivation differs in method and scope: it is entirely intersection-theoretic, built on Chow groups, blow-ups, and refined intersection products, hence functorial and valid for arbitrary closed subschemes. The subtlety for non-integral schemes is illustrated explicitly: for $\mathcal{I}_i$03 cut out by $\mathcal{I}_i$04, the bidegrees sum to $\mathcal{I}_i$05 while the subscheme of $\mathcal{I}_i$06 defined by the same equations has degree $\mathcal{I}_i$07 — so naive equation transfer fails, and the correct statement must be formulated through the correspondence $\mathcal{I}_i$08 rather than through shared defining ideals.

Limitations and open questions

Several hypotheses bear directly on the results. The graph-closure identification (Proposition 3.1) requires $\mathcal{I}_i$09 to be integral; the paper does not extend the isomorphism $\mathcal{I}_i$10 to non-integral bases, where the closure may acquire non-reduced structure differing from the blow-up. The multidegree formula is established for biprojective spaces with the multiprojective case asserted in a remark without full proof. The comparison of exceptional divisors relies on generic-point arguments with isomorphic local rings; extending these cycle-level equalities to finer invariants (e.g., refined Gysin pullbacks or operational classes) is not addressed. Finally, the degree formula concerns the specific correspondence $\mathcal{I}_i$11; characterizing when other natural correspondences between multidegrees and degrees hold remains open.

Conclusion

The paper provides a unified blow-up framework connecting simultaneous multigraded blow-ups, iterated blow-ups, and graph closures of rational maps to multiprojective spaces. Its two principal applications remove standing hypotheses from classical results: the product formula for Segre classes now holds for arbitrary closed embeddings without pure-dimensionality assumptions, and van der Waerden's degree formula is rederived functorially and extended from integral varieties to arbitrary closed subschemes of multiprojective spaces. The methods are elementary within Fulton's intersection theory, relying on universal properties of blow-ups, generic-point comparisons, and projection formulas, making the results accessible and broadly applicable wherever Segre classes and multidegrees appear.