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

Technical Summary: Blow-Up Constructions and Applications to Segre Classes and Multidegree Formulas

Problem Statement
The paper addresses two primary challenges in intersection theory concerning the behavior of blow-ups and their applications to characteristic classes and degree formulas. First, it seeks to establish a rigorous geometric relationship between simultaneous multigraded blow-ups and iterated blow-up constructions, specifically regarding the correspondence of their exceptional divisors. Second, it aims to derive intersection-theoretic formulas for the Chern classes of exceptional divisors arising from rational maps to multiprojective spaces.

A specific motivation for the first part is the proof of the product formula for Segre classes. Existing proofs in the literature (e.g., Fulton's Intersection Theory) generally rely on the hypothesis that the schemes involved are pure-dimensional to apply standard cycle-theoretic relations. The paper aims to remove this restriction. Similarly, for the second part, the paper addresses the need for a degree formula for arbitrary closed subschemes of multiprojective spaces, extending the classical result of van der Waerden, which was originally formulated for integral varieties using classical algebraic geometry.

Methodology
The author employs the formalism of Chow groups, refined intersection products, and the theory of blow-ups as developed by Fulton. The methodology is divided into two main sections:

1. Comparison of Blow-Up Constructions:
   The paper constructs a birational correspondence between two models:

   • The simultaneous blow-up of a product of schemes $Y_1 \times \dots \times Y_n \times \mathbb{A}^1_k$ along a specific closed embedding.
   • An iterated blow-up construction involving the product of exceptional divisors $E_1 \times \dots \times E_n$.
     By analyzing the normal cones and the projectivization of these cones, the author establishes a commutative diagram with Cartesian squares. This allows for the transfer of intersection-theoretic information between the two models, specifically relating the fundamental cycles of the exceptional divisors.

2. Graph Closure and Blow-Up Isomorphism:
   The paper investigates the relationship between the blow-up of a scheme $X$ along an ideal sheaf generated by global sections and the scheme-theoretic closure of the graph of the associated rational map to a multiprojective space.

   • It identifies the blow-up with the relative Proj of the Rees algebra.
   • It demonstrates that for integral schemes, the graph closure is isomorphic to the blow-up.
   • This framework is applied to the specific case of $\mathbb{P}^{m+n+1}_k$ and sections defining a map to $\mathbb{P}^m_k \times \mathbb{P}^n_k$.
   • Using the "Miracle Flatness" theorem, the author analyzes the fibers of the resulting morphism (isomorphic to $\mathbb{P}^1$) to derive a relation between the Chern classes of the pullback of tautological line bundles and the exceptional divisor.

Key Contributions and Results

1. Birational Correspondence of Exceptional Divisors:
   The paper establishes the existence of morphisms $\phi$ and $\varphi$ that create a commutative diagram between the iterated blow-up model and the simultaneous blow-up model. Crucially, it proves that the pullback of the fundamental cycle of the exceptional divisor in the iterated model equals the fundamental cycle of the exceptional divisor in the simultaneous model ($\phi^*[E'] = [E]$).

2. Product Formula for Segre Classes (General Case):
   Using the birational correspondence, the paper provides a proof of the product formula for Segre classes:
   $$s(X_1 \times \dots \times X_n, Y_1 \times \dots \times Y_n) = s(X_1, Y_1) \times \dots \times s(X_n, Y_n)$$
   Significance: Unlike previous proofs, this derivation does not require the schemes $X_i$ or $Y_i$ to be pure-dimensional. The proof reduces the problem to the case of Cartier divisors on a suitable resolution via the exceptional divisor, which acts as a geometric bridge.

3. Multidegree Formula for Arbitrary Subschemes:
   The paper derives a formula relating the codimension-$d$ degree of a closed subscheme $V' \subset \mathbb{P}^{m+n+1}_k$ to the codimension-$d$ bidegrees of a closed subscheme $V \subset \mathbb{P}^m_k \times \mathbb{P}^n_k$.

   • Result: The codimension-$d$ degree of $V'$ is equal to the sum of the codimension-$d$ bidegrees of $V$.
   • Significance: This result recovers and extends the classical degree formula of van der Waerden. While van der Waerden's original argument relied on the assumption that varieties were integral, this proof is entirely intersection-theoretic and applies to arbitrary closed subschemes, including non-integral ones. The paper notes that for non-integral subschemes, the relationship between degrees and bidegrees is more subtle than in the integral case.

Significance and Claims
The paper claims that its constructions provide a "geometric mechanism" for transferring intersection-theoretic information, thereby offering a conceptual and functorial derivation of the product formula for Segre classes and the multidegree formula.

• Regarding Segre Classes: The work demonstrates that the product formula is "fundamentally a consequence of the geometry of blow-ups," allowing for the removal of the pure-dimensionality hypothesis that constrained previous proofs.
• Regarding Degree Formulas: The approach offers a modern reformulation of van der Waerden's formula, extending its validity from integral varieties to arbitrary closed subschemes within the framework of Chow groups and Fulton's intersection theory.

The paper does not propose experimental applications or future implications beyond these theoretical extensions within intersection theory. It strictly adheres to the conventions and notation of Fulton's Intersection Theory [Ful98].