Minimization of hypersurfaces

Abstract

Let F ∈ Z[x 0 , . . . , x n ] be homogeneous of degree d and assume that F is not a 'nullform', i.e., there is an invariant I of forms of degree d in n + 1 variables such that I(F) = 0. Equivalently, F is semistable in the sense of Geometric Invariant Theory. Minimizing F at a prime p means to produce T ∈ Mat(n+1, Z)∩GL(n+1, Q) and e ∈ Z ≥0 such that F 1 = p −e F([x 0 , . . . , x n ] • T ) has integral coefficients and v p (I(F 1 )) is minimal among all such F 1 . Following Kollár [Kol97], the minimization process can be described in terms of applying weight vectors w ∈ Z n+1 ≥0 to F. We show that for any dimension n and degree d, there is a complete set of weight vectors consisting of [0, w 1 , w 2 , . . . , w n ]This answers a question raised by Kollár.Based on this result and a further study of the minimization process in the planar case n = 2, we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree d. We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. The algorithms are available in Magma.