2026/07/27 by Seung-Jo Jung, Morihiko Saito
#math.AG
Let X⊂\mathbb Pn-1 be a hypersurface of degree d≥3 with ordinary double points, where n≥3. 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∈\mathbb Z∩[n,nd-n-pf] with pf a positive integer. Here pf is bounded above by the minimal positive integer qs satisfying \binomqs+n-1n-1>s:=|\rm Sing X|, and we can verify that pf coincides with qs 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-2n-1≥ s or \binomd+n-3n-1≥ 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 A2-singularities instead of ordinary double points.