[Paper Review] Totally invariant divisors of endomorphisms of projective spaces

Hoering, Andreas|arXiv (Cornell University)|Jan 24, 2016

Algebraic Geometry and Number Theory15 references4 citations

TL;DR

This paper establishes a lower bound for the degree of the non-normal locus of totally invariant prime divisors under endomorphisms of projective space, using logarithmic differentials and Chern class computations. It proves that in P³, any totally invariant prime divisor must be a hyperplane, confirming the linearity conjecture in this case and showing that divisors of degree n are not totally invariant due to singularity constraints.

ABSTRACT

Totally invariant divisors of endomorphisms of the projective space are expected to be always unions of linear spaces. Using logarithmic differentials we establish a lower bound for the degree of the non-normal locus of a totally invariant divisor. As a consequence we prove the linearity of totally invariant divisors for $P^3$.

Motivation & Objective

• To resolve the conjecture that totally invariant subvarieties of endomorphisms f: Pⁿ → Pⁿ are always linear subspaces, particularly for divisors.
• To understand the geometric and cohomological constraints on totally invariant prime divisors in projective space.
• To prove that in P³, any totally invariant prime divisor must be a hyperplane, thereby confirming the linearity conjecture in dimension 3.
• To establish a sharp lower bound on the degree of the non-normal locus of such divisors, showing that high-degree divisors cannot be totally invariant unless they are singular in a controlled way.

Proposed method

• Uses the logarithmic cotangent sheaf ΩPⁿ(log D) to analyze the singularities of a totally invariant prime divisor D.
• Applies the logarithmic ramification formula to relate the canonical divisor of Pⁿ to the pullback of the divisor and a ramification divisor R.
• Computes the first and second Chern classes of ΩPⁿ(log D) ⊗ OPⁿ(1) using the residue exact sequence and the normalisation map ν: ˜D → D.
• Leverages Bott’s theorem and global sections of ΩPⁿ(log D) ⊗ OPⁿ(1) to construct a morphism from O⊕ⁿ⁺¹Pⁿ to the logarithmic bundle.
• Applies a key lemma on Chern classes of sheaf morphisms to compare c₂ of the pullback and the original bundle on a general surface S = f⁻¹(P²).
• Uses the fact that f is polarized (f*H ≡ mH) and iterates f to ensure invariance of the non-normal locus Z, enabling comparison of Chern classes via degree arguments.

Experimental results

Research questions

• RQ1 Can a totally invariant prime divisor of degree d ≥ 2 in Pⁿ be non-linear, particularly in P³?
• RQ2 What constraints do logarithmic differentials and Chern classes impose on the non-normal locus of such divisors?
• RQ3 Is it possible for a hypersurface of degree d = n to be totally invariant under an endomorphism of Pⁿ?
• RQ4 Does the non-normal locus of a totally invariant divisor have a degree bounded below by a function of n and d?
• RQ5 Can the linearity conjecture for totally invariant subvarieties be confirmed in P³?

Key findings

• The degree of the non-normal locus Z of a totally invariant prime divisor D ⊂ Pⁿ of degree d ≥ 2 satisfies deg(Z) > (d − 1)² − n(n − 1)/2.
• In P³, any totally invariant prime divisor must be a hyperplane, as the inequality forces deg(Z) > 1, but for d = 4, the upper bound on deg(Z) for plane curves is 1, leading to a contradiction unless D is smooth or linear.
• For d = n, the inequality deg(Z) > ½(n − 2)(n − 1) contradicts the known upper bound deg(Z) ≤ ½(n − 1)(n − 2) for irreducible plane curves, proving that no irreducible divisor of degree n can be totally invariant.
• The result improves upon previous work by showing that singular normal hypersurfaces of degree n do not admit endomorphisms induced from Pⁿ.
• The strict inequality in the degree bound is essential: equality would lead to a contradiction in the asymptotic behavior of Chern class terms, proving the bound is sharp.
• The method confirms the linearity conjecture in P³ and provides a general framework for studying totally invariant subvarieties via logarithmic geometry and sheaf-theoretic invariants.