Polyhedral Divisors of Affine Trinomial Hypersurfaces

Abstract

We find polyhedral divisors corresponding to the torus action of complexity one on affine trinomial hypersurfaces. Explicit computations for particular classes of such hypersurfaces including Pham-Brieskorn surfaces and rational trinomial hypersurfaces are given.

The Nash problem for torus actions of complexity one

“…Using the work of Kruglov (cf. [Kru19]), which makes the Altmann-Hausen description explicit for trinomial hypersurfaces, our main result on the Nash problem implies the following.…”

“…For z ∈ P 1 \ {0, 1, ∞} set D z = σ. Then D := z∈P 1 D z • {z} is a p-divisors with locus P 1 , and X endowed the G m -action α • (x 0 , x 1 , x 2 ) = (α 20 x 0 , α 15 x 1 , α 12 x 2 ) is isomorphic to X(D) (see [Kru19] for more general computations of polyhedral divisors defining affine trinomial hypersurfaces) One easily checks that Min(DV(X) But on the toroidification, the divisor corresponding to [•, 1, 0] is a (−1)-curve. Contracting this curve, one obtains the minimal resolution of X, which is an equivariant resolution of X which does not factor through the toroidification.…”