允许具有Picard数一的Fano三维簇的$\mathbb Q$-Gorenstein光滑化的加权射影空间
Weighted projective spaces admitting $\mathbb Q$-Gorenstein smoothings to Fano threefolds with Picard number one
- National Taiwan University（国立台湾大学）
- Center for Complex Geometry, Institute for Basic Science (IBS)（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总结：
研究加权射影三维簇允许Q-戈伦斯坦光滑化为Picard数一的光滑Fano三维簇，推导必要数值与局部条件，计算机搜索确定数值候选，构造若干光滑化并证明一个非光滑化例子，给出消失环准则保持Picard数一。
AI中文摘要：
我们研究良形的加权射影三维簇$X=\mathbb P(w_0,w_1,w_2,w_3)$，这些簇允许$\mathbb Q$-Gorenstein光滑化为Picard数为一的光滑Fano三维簇。对于del Pezzo三维簇$V_d$和素数Fano三维簇$Y_g$，我们推导出三个必要的数值和局部条件：反规范体积方程、由反规范Hilbert多项式的线性项得到的恒等式，以及沿坐标曲线光滑化横截$A$-奇点的全局截面条件。利用这些条件进行的计算机搜索确定了所有数值候选，除了已知的$V_5$的无穷族，其中$w_0+w_1+w_2+w_3\leq 5000$。我们从加权射影三维簇的数值候选构造了到$V_1, V_2, Y_6, Y_{10},$和$Y_{12}$的$\mathbb Q$-Gorenstein光滑化，并证明了$\mathbb P(2,5,8,25)$虽然是$V_4$的数值候选，但不是$\mathbb Q$-Gorenstein可光滑化的。为了识别光滑纤维，我们还给出了一个消失环准则，确保在光滑化下Picard数一得以保持。
英文摘要：
We study well-formed weighted projective threefolds $X=\mathbb P(w_0,w_1,w_2,w_3)$ that admit a $\mathbb Q$-Gorenstein smoothing to a smooth Fano threefold of Picard number one. For del Pezzo threefolds $V_d$ and prime Fano threefolds $Y_g$, we derive three necessary numerical and local conditions: the anticanonical volume equation, an identity obtained from the linear term of the anticanonical Hilbert polynomial, and a global section condition for smoothing the transversal $A$-singularities along coordinate curves. A computer search using these conditions determines all numerical candidates, apart from the known infinite family for $V_5$, with $w_0+w_1+w_2+w_3\leq 5000$. We construct $\mathbb Q$-Gorenstein smoothings to $V_1, V_2, Y_6, Y_{10},$ and $Y_{12}$ from numerical candidates of weighted projective threefolds, and prove that $\mathbb P(2,5,8,25)$, although a numerical candidate for $V_4$, is not $\mathbb Q$-Gorenstein smoothable. To identify the smooth fibers, we also give a vanishing-cycle criterion ensuring that Picard number one is preserved under the smoothing.
