具有普通双点的射影超曲面的伯恩斯坦 - 佐藤多项式的根
Roots of Bernstein-Sato polynomials for projective hypersurfaces with ordinary double points
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中文总结 AI 辅助
研究具有普通双点的射影超曲面的伯恩斯坦 - 佐藤多项式的根，给出根的相关形式及\(p_f\)的上界，通过构造齐次多项式证明特定情况下上界精确，并提及相关猜想。
中文摘要 AI 辅助
设\(X\subset{\mathbb P}^{n - 1}\)是次数\(d\geq3\)且具有普通双点的超曲面，其中\(n\geq3\)。其定义多项式\(f\)的伯恩斯坦 - 佐藤多项式的根在符号上由\(1\)，\((n - 1)/2\)以及\(j/d\)给出，其中\(j\in{\mathbb Z}\cap[n,nd - n - p_f]\)，\(p_f\)是正整数。\(p_f\)有上界，由满足\(\binom{q_s + n - 1}{n - 1}>s := |{\rm Sing}\,X|\)的最小正整数\(q_s\)给出，且当\(X\)的奇点处于“一般位置”时\(p_f\)与\(q_s\)重合。通过给出次数为\(d\)的齐次多项式表明在特定情况下该上界是精确的。还提到关于该精确界的一个猜想。
英文摘要
Let $X\subset{\mathbb P}^{n-1}$ be a hypersurface of degree $d\ge3$ with ordinary double points, where $n\ge3$. The roots of Bernstein-Sato polynomial of its defining polynomial $f$ are given up to sign by 1, $(n-1)/2$, and $j/d$ for $j\in{\mathbb Z}\cap[n,nd-n-p_f]$ with $p_f$ a positive integer. Here $p_f$ is bounded above by the minimal positive integer $q_s$ satisfying $\binom{q_s+n-1}{n-1}>s:=|{\rm Sing}\,X|$, and we can verify that $p_f$ coincides with $q_s$ in the case the singular points of $X$ are in ``general position". We show that this upper bound is sharp in the case $\binom{\lfloor d/2\rfloor+n-2}{n-1}\ge s$ or $\binom{d+n-3}{n-1}\ge sn$ by providing a homogeneous polynomial of degree $d$ such that the associated projective hypersurface has ordinary double points at given $s$ points in sufficiently general position and is nonsingular outside them (using a theorem of Alexander and Hirschowitz for the second case). It is conjectured that the above sharp bound under the first hypothesis can be extended naturally to the case where $X$ has only $A_2$-singularities instead of ordinary double points.
