本文研究实光滑射影三维簇有理曲线纤维化的拓扑，证明几何有理底曲面时 Seifert 纤维化或透镜空间连通和的参数 k 不超过 4，并肯定回答 Kollár 的两个问题，通过实奇异 Del Pezzo 曲面分析得出。
摘要：
设 W -> X 为一个实光滑射影三维簇，其纤维为有理曲线。Kollár 证明了若 W(R) 是可定向的，则 W(R) 的一个连通分量 N 本质上要么是 Seifert 纤维化流形，要么是透镜空间的连通和。设 k := k(N) 为如下定义的整数：若 g : N -> F 是 Seifert 纤维化，则定义 k := k(N) 为 g 的多重纤维数；若 N 是透镜空间的连通和，则 k 定义为不同于 P^3(R) 的透镜空间数。我们的主定理断言：若 X 是几何有理曲面，则 k <= 4。此外，我们证明若 F 微分同胚于 S^1xS^1，则 W(R) 是连通的且 k = 0。这些结果肯定地回答了 Kollár 的两个问题，他在 1999 年证明了 k <= 6 并提出 4 可能是最佳上界。我们通过对仅具有 Du Val 奇点的实奇异 Del Pezzo 曲面的细致研究推导出该定理。
英文摘要：
Let W -> X be a real smooth projective threefold fibred by rational curves. Kollár proved that if W(R) is orientable a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Let k : = k(N) be the integer defined as follows: If g : N -> F is a Seifert fibration, one defines k : = k(N) as the number of multiple fibres of g, while, if N is a connected sum of lens spaces, k is defined as the number of lens spaces different from P^3(R). Our Main Theorem says: If X is a geometrically rational surface, then k <= 4. Moreover we show that if F is diffeomorphic to S^1xS^1, then W(R) is connected and k = 0. These results answer in the affirmative two questions of Kollár who proved in 1999 that k <= 6 and suggested that 4 would be the sharp bound. We derive the Theorem from a careful study of real singular Del Pezzo surfaces with only Du Val singularities.
We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type A the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties.
We classify RDP del Pezzo surfaces with global vector fields over arbitrary algebraically closed fields of characteristic $p \neq 2$. In characteristic $0$, every RDP del Pezzo surface $X$ is equivariant, that is, ${\rm Aut}_X = {\rm Aut}_{\widetilde{X}}$, where $\widetilde{X}$ is the minimal resolution of $X$, hence the classification of RDP del Pezzo surfaces with global vector fields is equivalent to the classification of weak del Pezzo surfaces with global vector fields. In this article, we show that if $p \neq 2,3,5,7$, then it is still true that every RDP del Pezzo surface is equivariant…
In this paper we classify nodal rational non-$\mathbb{Q}$-factorial del Pezzo threefolds of degree 2 which can be $G$-birationally rigid for some subgroup $G\subset \operatorname{Aut}(X)$.
Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$-condition are birationally rigid: their Mori fibre space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for "most" families a general de…
本文首先给出了Picard数1 Gorenstein log del Pezzo曲面的奇异类型的完整列表以及与给定奇异类型同构的类的个数。然后给出了一种求Gorenstein log del Pezzo曲面所有奇异类型的方法。作为一个应用，我们提出了完整的列表的Dynkin型的Picard数2 Gorenstein日志del Pezzo表面。最后给出了相对极小Gorenstein log del Pezzo曲面的奇异类型的完整列表以及与给定奇异类型同构的类的个数。
