环面Fano簇的积分与有理Gongyo指数之间的差距
On the gap between the integral and rational Gongyo indices of toric Fano varieties
- Fukuoka University（福冈大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究环面Fano簇的有理与积分Gongyo指数之差，证明维数下界并确定等号情形，构造无穷正差距族，并证明所有正有理数均可作为差距出现。
AI中文摘要：
对维数至多为7的光滑环面Fano簇的精确计算表明，有理Gongyo指数与积分Gongyo指数之间的正差存在一个依赖于维数的下界。我们通过反典范极小模型程序，计算了由射影空间在环面不变点处爆破得到的光滑环面Fano模型的两个指数。对于伪对称光滑环面Fano簇，我们证明了所提出的下界，并确定了所有等号情形。我们还应用一个升维构造，从具有零差距的奇数维弱Fano簇获得一个显式的无穷族光滑环面Fano簇，其差距为正。对于所考虑的标准模型，我们确定了通过迭代此构造获得Fano簇所需的最小维数增加，并计算了最小提升的两个指数。最后，每个正有理数都作为Gongyo指数差距出现，即使在伪对称类中也是如此。
英文摘要：
Exact calculations for smooth toric Fano varieties in dimensions at most seven suggest a sharp dimension-dependent lower bound for the positive difference between the rational and integral Gongyo indices. We compute both indices for the smooth toric Fano models obtained from blow-ups of projective space at torus-invariant points by the anticanonical minimal model program. For pseudo-symmetric smooth toric Fano varieties, we prove the proposed lower bound and determine all equality cases. We also apply a dimension-raising construction to obtain an explicit infinite family of smooth toric Fano varieties with positive gap from odd-dimensional weak Fano varieties with zero gap. For the standard models considered, we determine the smallest dimension increase needed to obtain a Fano variety by iterating this construction and compute both indices of the minimal lifts. Finally, every positive rational number occurs as a Gongyo-index gap, even within the pseudo-symmetric class.
