On rigidity of factorial trinomial hypersurfaces

Abstract

Abstract. An affine algebraic variety X is rigid if the algebra of regular functions K[X] admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least 2.

The automorphism group of a rigid affine variety

“…Consider a linear combination i, j c i j u i j with integer coefficients in the lattice M. Since M is the factor group of the group K by its torsion part, we have i, j c i j u i j = 0 if and only if there exists a positive integer m such that m i, j c i j w i j = 0 in the group K or, equivalently, m i, j c i j e i j = a 1 ⎛ ⎝ l 01 e 01 − n 1 j=1 l 1 j e 1 j ⎞ ⎠ + a 2 ⎛ ⎝ l 01 e 01 − n 2 j=1 l 2 j e 2 j ⎞ ⎠ in the lattice Z n for some integers a 1 and a 2 . This shows that there are exactly two ways to express the weight ν = l 01 u 01 as a linear combination of elements of a proper subset of the set of extremal weights, and the coefficients of these combinations are precisely the exponents of the monomials T l 1 1 and T l 2 2 respectively. Case 3.…”

“…By [, Theorem 1.1], a factorial trinomial hypersurface X is rigid if and only if every exponent in the trinomial f is greater or equal to 2. This fact together with Theorem implies the following result.…”

“…Examples of rigid trinomial hypersurfaces can be found in , , . In [, Theorem 1.1], a criterion for rigidity of a factorial trinomial hypersurface is obtained. It turns out that many trinomial hypersurfaces are rigid.…”

Polyhedral Divisors of Affine Trinomial Hypersurfaces

“…The case of a single trinomial equation leads to affine trinomial hypersurfaces. The problem of computing pp-divisors corresponding to the action of a torus on a trinomial hypersurface was previously discussed in [3]. Namely, Arzhantsev has found pp-divisors corresponding to the action of a torus on factorial trinomial hypersurfaces.…”

Polynomial curves on trinomial hypersurfaces

“…In 1873, Schwartz [20] found polynomial solutions of equation (1) in coprime polynomials z 0 (x), z 1 (x), z 2 (x) for every platonic triple (p, q, r) with p, q, r ≥ 2; see also [22] and [8] for explicit formulas.…”