Theorems of Bertini and Chevalley

Technical Summary: Theorems of Bertini and Chevalley

Problem Statement
The paper addresses two fundamental results in algebraic geometry: Bertini's irreducibility theorem and Chevalley's structure theorem for algebraic groups.

1. Bertini's Irreducibility Theorem: The problem is to establish conditions under which the inverse image of a general hyperplane under a dominant morphism remains geometrically irreducible. Specifically, for a geometrically irreducible variety $X$ over an infinite field $k$ and a dominant morphism $\pi: X \dashrightarrow \mathbb{P}^n$ ($n \geq 2$), one must prove that $\pi^{-1}(H)$ is geometrically irreducible for a general hyperplane $H$.
2. Chevalley's Theorem: The problem is to characterize the structure of a connected, smooth algebraic group $G$ over a perfect field $k$. The theorem asserts that $G$ fits into an exact sequence $1 \to L \to G \to A \to 0$, where $L$ is an affine algebraic group and $A$ is a proper algebraic group (an Abelian variety).

Methodology
The author, János Kollár, aims to provide proofs that rely minimally on advanced machinery, specifically restricting the use of results to those found in Shafarevich's textbook [Sha74], the existence of Jacobians for smooth projective curves, and Weil's theorems on rational maps to algebraic groups.

• For Chevalley's Theorem:

  • Rational Actions and Weil Divisors: Instead of requiring a regular (everywhere defined) action of $G$ on a projective variety, the proof utilizes a rational action on a normal projective variety $X$. This necessitates working with Weil divisors rather than just Cartier divisors.
  • Morphisms to Jacobians: For a Weil divisor $D$ on $X$, the author constructs a morphism $\alpha_{X,D}: G \to \prod \text{Jac}(C_i)$, where $C_i$ are smooth curves in $X$. This map sends $g \in G$ to the linear equivalence class of the restriction of the translated divisor $D_g$ to the curves.
  • Kernel Analysis: The proof identifies a subgroup $H \subset G$ (the identity component of the kernel of $\alpha_{X,D}$) and demonstrates that $H$ is affine. This is achieved by constructing a linear representation of $H$ on a carefully chosen linear subsystem of divisors.
  • Handling Singularities: A key technical step involves choosing a divisor $D$ with an isolated singular point. The stabilizer of such a divisor is shown to be finite, ensuring the representation is faithful and the kernel is affine. The proof navigates subtle issues regarding the birational transform of linear systems (Warning 7.4.1) by restricting to geometrically irreducible parameter sets.

• For Bertini's Irreducibility Theorem:

  • Reduction to Pencils: The proof reduces the general case to the study of pencils of divisors.
  • Base Locus Analysis: It utilizes the properties of the base locus of a pencil. If the base locus is smooth and of codimension 2, and members are smooth at a point, general members are geometrically irreducible.
  • Frobenius and Smoothness: In cases where the morphism is not generically smooth, the proof employs the Frobenius morphism to reduce to a generically smooth situation.
  • Induction: The result is extended to complete intersections via induction, utilizing a stronger version of the theorem (Corollary 24) that handles linear systems that are not "fully general" but satisfy specific codimension conditions on singular loci.

Key Contributions and Results

1. Shortened Proofs: The paper provides a proof of Bertini's irreducibility theorem that is claimed to be shorter than existing proofs in the literature, relying on elementary algebraic geometry tools (specifically from [Sha74]) and the Severi-Weil lemma.
2. Chevalley's Theorem via Weil Divisors: The proof of Chevalley's theorem is reformulated to explicitly use Weil divisors and rational actions. This approach avoids the need for more complex compactification theorems (like Nagata's or Weil's regularization) that might otherwise be required to ensure projectivity or regularity of the action.
3. Severi-Weil Lemma: The paper reiterates and utilizes the Severi-Weil lemma, which states that linear equivalence of divisors on a variety can be checked on a finite set of smooth curves. This is central to constructing the morphism from the algebraic group to the product of Jacobians.
4. Technical Lemmas:
   • Lemma 11: Establishes that any rational homomorphism of algebraic groups is a regular homomorphism.
   • Lemma 12: Shows how rational actions on linear systems induce regular linear representations.
   • Corollary 19: Clarifies that for geometrically irreducible spanning sets, the birational transform of the linear system behaves as expected (i.e., the transform of the system is spanned by the transforms of general members).

Significance and Claims
The paper claims that the primary significance of this work is methodological: it demonstrates that one "should not be afraid of working with Weil divisors." The author argues that Weil divisors provide an "optimal framework" for certain questions, particularly in the context of Chevalley's theorem, where they allow for a more direct proof using rational actions without requiring the full strength of modern compactification theorems.

The author explicitly states that the proofs are designed to be self-contained regarding the use of advanced machinery, relying only on the existence of Jacobians and basic results from Shafarevich. The paper does not claim to discover new theorems but rather to revisit and streamline the proofs of these well-known results, offering a perspective that emphasizes the utility of Weil divisors and rational maps in the study of algebraic groups and irreducibility.