AI 中文总结
该研究针对环面点爆破的Cox环，给出Rees重数1生成的有限判据，将其应用于伪加权射影空间与射影环面曲面，得到重数1生成的等价条件。
AI 中文摘要
设X为完全环面簇，e为其开环面的单位元。Bl_eX的Cox环由环面点格理想I_X的饱和幂描述。我们基于投影格理想的解析展布及相伴分次环中的零因子，给出Rees重数1生成的有限判据。首先将该判据应用于伪加权射影空间，证明重数1生成等价于I_X为完全交；对射影环面曲面，证明重数1生成成立当且仅当被X支配的每个皮卡数1的环面曲面均满足该性质。
英文摘要
Let $X$ be a complete toric variety and let $e$ be the identity of its open torus. The Cox ring of ${\rm Bl}_eX$ is described by saturated powers of the toric-point lattice ideal $I_X$. We give a finite criterion for generation in Rees multiplicity one in terms of analytic spreads of projected lattice ideals and zero divisors in the associated graded ring. We apply this criterion first to fake weighted projective spaces, proving that multiplicity-one generation is equivalent to $I_X$ being a complete intersection. For projective toric surfaces, we prove that multiplicity-one generation holds if and only if it holds for every Picard-number-one toric surface dominated by $X$.
