Sharp FRS 与离对角 Gram 映射的奇点阈值
Sharp FRS and singularity thresholds for off-diagonal Gram maps
- Kennesaw State University（肯尼索州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文确定了离对角 Gram 映射具有平坦有理奇点的精确阈值，并给出了零纤维理想的 log-canonical 阈值，通过加权估计与各向同性下界证明，同时提供了局部域上的连续性模。
AI中文摘要：
在特征为零的 M 维二次空间中，h 个向量的离对角 Gram 映射具有平坦且纤维约化的有理奇点（FRS），当且仅当 M > max_{2 <= r <= h} (2h - r + 1 - 2h/r)。等价地，这给出了映射 x |-> (x_i x_j)_{i < j} 的精确 FRS 卷积阈值。对于其零纤维理想 I_{M,h}，我们还证明了 lct(I_{M,h}) = min{ binom(h,2), min_{0 <= k <= h-2} ( binom(k+1,2) + M(h-k)/2 ) }。在严格范围内，所有截断系数映射都具有几何整的完全交纤维，并且在 Z[1/2] 上忠实平坦，且在 h 个顶点的图上一致成立。证明使用了有限局部主理想环上空心对称矩阵的尖锐加权估计，并结合各向同性下界。在已有的有理奇点点计数方法框架内，该估计还为每个奇剩余特征局部域上的 Gram 映射密度给出了显式的连续性模。
英文摘要：
The off-diagonal Gram map for h vectors in an M-dimensional quadratic space is flat with reduced fibers of rational singularities (FRS) in characteristic zero if and only if M > max_{2 <= r <= h} (2h - r + 1 - 2h/r). Equivalently, this gives the exact FRS convolution threshold of x |-> (x_i x_j){i < j}. For its zero-fiber ideal I{M,h}, we also prove lct(I_{M,h}) = min{ binom(h,2), min_{0 <= k <= h-2} ( binom(k+1,2) + M(h-k)/2 ) }. In the strict range, all truncated coefficient maps have geometrically integral complete-intersection fibers and are faithfully flat over Z[1/2], uniformly over graphs on h vertices. The proof uses a sharp weighted estimate for hollow symmetric matrices over finite local principal ideal rings, paired with isotropic lower bounds. Within the established point-counting approach to rational singularities, this estimate also gives an explicit modulus of continuity for Gram-map densities over every local field of odd residue characteristic.
