Minimization of hypersurfaces

Abstract

Let $F\in \mathbb{Z}[x_0,\dots ,x_n]$ 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 $I(F)\ne 0$. Equivalently, $F$ is semistable in the sense of Geometric Invariant Theory. Minimizing $F$ at a prime $p$ means to produce $T\in \mathrm{Mat}(n+1,\mathbb{Z})\cap \mathrm{GL}(n+1,\mathbb{Q})$ and $e\in {\mathbb{Z}}_{\ge 0}$ such that $F_1=p^{-e}F([x_0,\dots ,x_n]\cdot T)$ has integral coefficients and $v_p(I(F_1))$ is minimal among all such $F_1$. Following Kollár, the minimization process can be described in terms of applying weight vectors $w\in {\mathbb{Z}}_{\ge 0}^{n+1}$ to $F$. We show that for any dimension $n$ and degree $d$, there is a complete set of weight vectors consisting of $[0,w_1,w_2,\dots ,w_n]$ with $0\le w_1\le w_2\le \cdots \le w_n\le 2n{d}^{n-1}$. When $n=2$, we improve the bound to $d$. This answers a question raised by Kollár. These results are valid in a more general context, replacing $\mathbb{Z}$ and $p$ by a PID $R$ and a prime element of $R$. 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 the computer algebra system Magma.

43 pages, various figures. v2: Added proof that there is always a unique minimal complete system of weight vectors, included referees' suggestions, fixed two mistakes. v3: Further edits following referee's report

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