Critical loci of self-maps of projective space

2026/07/24 by Matt Olechnowicz, Max Weinreich
#math.AG #math.DS

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Abstract

Let K be an algebraically closed field and let n, d ≥ 2. We show that the critical scheme of a general endomorphism of ℙⁿ_K of degree d is an integral hypersurface. This extends a result of Ingram--Ramadas--Silverman to arbitrary characteristic. We use basic facts about polynomial rings to show that certain carefully chosen examples have absolutely irreducible Jacobian. To handle the wild case where the characteristic of K divides d, we introduce a polynomial that imitates the homogeneous Jacobian determinant.

Citations

• On ramification of self-maps of P2

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