发表机构
National Taiwan University; Center for Complex Geometry, Institute for Basic Science (IBS)（台湾大学; 基础科学研究院复杂几何中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究允许到 $\mathbb P^3$ 的 $\mathbb Q$-Gorenstein 光滑化的加权射影三维簇，推导必要条件和半群条件，证明分类猜想在若干情形下成立，并给出有限性结果。
AI 中文摘要
我们研究允许到 $\mathbb P^3$ 的 $\mathbb Q$-Gorenstein 光滑化的良态加权射影三维簇。已知有两个族：$\mathbb P^2$ 型和 $Q$ 型，且猜想这些是仅有的可能性。我们推导了此类光滑化的数值和局部必要条件。除了反典范体积方程外，当所有余维二奇点均为 $A$ 型时，反典范 Hilbert 多项式的恒定性产生另一个恒等式。我们还获得了一个半群条件，该条件控制沿具有横向 $A$ 型奇点的余维二曲线的全局光滑化方向的存在性。我们应用这些条件在若干情形下证明了预期的分类。特别地，对于每个固定的无平方因子整数 $d$，满足 $\gcd(a,b)=d$ 的 $\mathbb Q$-Gorenstein 可光滑化空间 $\mathbb P(1,a,b,c)$ 仅有有限多个。我们的方法将可能的权重简化为有限的精确计算；对于每个素数 $p\le100$，计算仅产生两个预期族的成员。最后，我们证明了当 $\gcd(a,b)=d$ 且 $a=d^2$ 时的分类。
英文摘要
We study well-formed weighted projective threefolds that admit $\mathbb Q$-Gorenstein smoothings to $\mathbb P^3$. Two families are known: the $\mathbb P^2$-type and the $Q$-type, and it is conjectured that these are the only possibilities. We derive numerical and local necessary conditions for such a smoothing. In addition to the anticanonical volume equation, constancy of the anticanonical Hilbert polynomial yields a further identity when all codimension two singularities are of $A$-type. We also obtain a semigroup condition governing the existence of global smoothing directions along codimension two curves with transverse $A$-type singularities. We apply these conditions to prove the expected classification in several cases. In particular, for every fixed square-free integer $d$, there are only finitely many $\mathbb Q$-Gorenstein smoothable spaces $\mathbb P(1,a,b,c)$ such that $\gcd(a,b)=d$. Our method reduces the possible weights to a finite exact computation; for every prime $p\le100$, the computation produces only members of the two expected families. Finally, we prove the classification when $\gcd(a,b)=d$ and $a=d^2$.
Comments13 pages
