[Paper Review] On a connectedness theorem of Debarre

Lucian Bădescu|ArXiv.org|Dec 15, 2008

Advanced Topics in Algebra7 references3 citations

TL;DR

This paper strengthens a connectedness theorem of Debarre for products of projective spaces by proving that under a slightly stronger dimension condition, the preimage of the diagonal in a morphism from a complete irreducible variety is G3—meaning all formal-rational functions extend to rational functions. The key result generalizes Grothendieck–Faltings' connectedness and extension theorems to products of projective spaces using global geometric methods and formal function theory.

ABSTRACT

Under a slightly stronger hypothesis, one improves a connectedness result of Debarre [D] for a product of two projective spaces in terms of the extension problem of formal-rational functions (see Theorems 1.3 and 1.4 of the introduction)

Motivation & Objective

• To extend Debarre's connectedness theorem for products of projective spaces to a stronger extension-theoretic property: the G3 condition on formal-rational functions.
• To generalize the Grothendieck–Faltings result on hyperplane sections being G3 to subvarieties in products of projective spaces.
• To investigate whether the dimension hypotheses in Debarre's original theorem can be tightened to ensure the G3 property.
• To establish conditions under which the diagonal of a subvariety in a product of projective spaces is G3 in the product space.
• To explore the Grothendieck–Lefschetz condition and its effective version in the context of subvarieties in products of projective spaces.

Proposed method

• Uses the join construction and formal function theory to analyze the extension of formal-rational functions along subvarieties.
• Applies global geometric methods instead of local cohomological techniques, differing from Grothendieck–Faltings' original approach.
• Employs results on formal functions from Section 2 and adapts a global proof of a Bertini-type theorem by Bonacini, Del Padrone, and Nesci.
• Relies on Debarre's connectivity theorems as a foundational input for proving the stronger G3 condition.
• Uses the canonical map α_{X,Y}: K(X) → K(X_{/Y}) to define and analyze the G3 property via isomorphism of k-algebras.
• Applies the Grothendieck–Lefschetz condition and its effective variant to subvarieties of products of projective spaces under codimension bounds.

Experimental results

Research questions

• RQ1 Can Debarre’s connectedness theorem for products of projective spaces be strengthened to ensure the G3 property for the preimage of the diagonal?
• RQ2 Is the dimension hypothesis in Debarre’s theorem tight enough to guarantee the G3 condition, or can it be relaxed?
• RQ3 Does the G3 property for subvarieties in products of projective spaces hold under codimension bounds analogous to those in the classical Grothendieck–Faltings theorem?
• RQ4 Under what conditions does the Grothendieck–Lefschetz condition Lef(X,Y) or its effective version Leff(X,Y) hold for diagonals in subvarieties of products of projective spaces?
• RQ5 Is there a uniform bound on codimension, in terms of coampleness of the ambient space, ensuring the G3 property for the diagonal of a subvariety?

Key findings

• Under the hypothesis that dim((p_J × p_J)(f(X))) > ∑_{i∈J} n_i + p - 1 for all non-empty J with |J| = p, the preimage f⁻¹(Δ) is G3 in X.
• For a closed irreducible subvariety Z ⊂ ℙ^{n₁} × ℙ^{n₂} with codim_P Z < ½ min{n₁, n₂}, the diagonal Δ_Z ⊂ Z × Z is G3 in Z × Z.
• The pair (Z × Z, Δ_Z) satisfies the Grothendieck–Lefschetz condition Lef(Z × Z, Δ_Z) if codim_P Z < ½ n₂ (assuming n₁ ≥ n₂ ≥ 1).
• The effective Grothendieck–Lefschetz condition Leff(Z × Z, Δ_Z) is never satisfied under the hypotheses of Corollary 4.7.
• The coampleness of ℙ^{n₁} × ℙ^{n₂} is n₂, and the G3 property for the diagonal holds when codim_P Z < ½ ca(P), generalizing known results for ℙ^n.
• The paper leaves open whether the dimension hypothesis in Theorem 1.3 can be weakened to match Debarre’s original condition, suggesting a potential direction for future work.