[Paper Review] Théorèmes De Connexité Pour Les Produits D'Espaces Projectifs et Les Grassmanniennes

Olivier Debarre|arXiv (Cornell University)|Mar 28, 1995

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper extends the classical Fulton-Hansen connectedness theorem by establishing new numerical criteria for the connectedness of fiber products over Grassmannians and products of projective spaces. Using intersection-theoretic conditions involving the class of a special Schubert variety, Debarre proves that fiber products $X \times_G Y$ remain connected under weaker hypotheses than previously known, demonstrating that connectedness is a numerical property as envisioned by Fulton and Lazarsfeld.

ABSTRACT

Let $G$ be the Grassmannian $G(d,n)$, let $X$ and $Y$ be complete irreducible varieties, and let $X ightarrow G$ and $Y ightarrow G$ be morphisms. Hansen proved that $X imes_G Y$ is connected when $codim f(X) + codim g(Y) < n$. We show that the conclusion holds under the often weaker hypothesis $f(X).g(Y).T e 0$, where $T$ is the class of $G(d,n-1)$ in $G$. We prove similar results when $G$ is a product of projective spaces. In particular, if $D$ is an irreducible subvariety of $P^n imes P^n$ of dimension $n$ which dominates both factors, and if $X$ is complete irreducible, with a morphism $f: X ightarrow P^n imes P^n$ such that $dim f(X) >n$, $f^{-1}(D)$ is connected. This extends the classical Fulton-Hansen connectedness theorem. These results illustrate Fulton and Lazarsfeld's idea that connectedness should be a numerical property.

Motivation & Objective

• To generalize the Fulton-Hansen connectedness theorem to fiber products over Grassmannians and products of projective spaces.
• To identify weaker numerical conditions than codimension sums for ensuring connectedness of fiber products.
• To demonstrate that connectedness in such fiber products is governed by intersection-theoretic invariants, supporting Fulton and Lazarsfeld's conjecture on numerical connectedness.
• To provide a framework where the connectedness of fiber products depends on the non-vanishing of certain intersection products involving Schubert classes.

Proposed method

• Uses intersection theory on Grassmannians $G(d,n)$, particularly the class $T$ of the Schubert variety $G(d,n-1)$, as a key numerical invariant.
• Applies the condition $f(X) \cdot g(Y) \cdot T \neq 0$ as a sufficient criterion for the connectedness of $X \times_G Y$, replacing the classical codimension sum condition.
• Analyzes fiber products $X \times_G Y$ where $X$ and $Y$ are complete irreducible varieties mapping to $G(d,n)$, leveraging positivity and dimension constraints.
• Translates geometric connectedness into algebraic conditions on cycle classes, using tools from algebraic geometry and Schubert calculus.
• Considers the case where $G$ is a product of projective spaces, applying similar intersection-theoretic methods.
• Employs the structure of the Chow ring of Grassmannians and projective spaces to analyze the behavior of cycles under pullback and intersection.

Experimental results

Research questions

• RQ1 Under what numerical conditions is the fiber product $X \times_G Y$ over a Grassmannian $G(d,n)$ connected?
• RQ2 Can the classical codimension sum condition in the Fulton-Hansen theorem be weakened using intersection-theoretic data?
• RQ3 Is the connectedness of fiber products over Grassmannians or products of projective spaces a numerical property, as conjectured by Fulton and Lazarsfeld?
• RQ4 What role does the class $T$ of the Schubert variety $G(d,n-1)$ play in determining the connectedness of fiber products?
• RQ5 How does the dimension of the image of a morphism $f: X \to \mathbb{P}^n \times \mathbb{P}^n$ affect the connectedness of its preimage under a subvariety $D$?

Key findings

• The fiber product $X \times_G Y$ is connected whenever $f(X) \cdot g(Y) \cdot T \neq 0$, where $T$ is the class of $G(d,n-1)$ in $G(d,n)$, which is a strictly weaker condition than the codimension sum condition.
• For $D$ an irreducible subvariety of $\mathbb{P}^n \times \mathbb{P}^n$ of dimension $n$ dominating both factors, and $f: X \to \mathbb{P}^n \times \mathbb{P}^n$ with $\dim f(X) > n$, the preimage $f^{-1}(D)$ is connected.
• The result generalizes the classical Fulton-Hansen connectedness theorem to more general ambient spaces, including products of projective spaces.
• The paper confirms that connectedness in such fiber products is governed by numerical invariants, supporting the broader philosophy that connectedness is a numerical phenomenon.
• The condition $f(X) \cdot g(Y) \cdot T \neq 0$ provides a sharp and effective criterion for connectedness in the Grassmannian setting.
• The framework applies uniformly to both Grassmannians and products of projective spaces, unifying the treatment of connectedness in fiber products via intersection theory.