[论文解读] A construction of smooth varieties admitting small contractions

Yuto Masamura, Tomoki Yoshida|arXiv (Cornell University)|Jan 20, 2026

Algebraic Geometry and Number Theory被引用 0

一句话总结

作者通过两次逐步吹胀从任意光滑投影簇构造具有小收缩的平滑簇，推广 Kawamata 的四维示例，并给出用于可数清晰度的分割线以判定在来自 del Pezzo 表面的积上形成的弱 Fano 四维的应用的可分性判据。

ABSTRACT

We construct smooth varieties admitting small contractions from arbitrary smooth projective varieties. This construction generalizes Kawamata's four-dimensional example. We also give sufficient conditions for divisors on these varieties to be nef. As an application, we obtain weak Fano fourfolds from products of two del Pezzo surfaces.

研究动机与目标

• Motivate the study of small contractions as fundamental birational phenomena in the MMP and the challenge of producing explicit smooth examples in dimension four or higher.
• Develop a general blowup construction that yields smooth varieties admitting small contractions with flexible centers and possibly positive-dimensional intersections.
• Provide criteria to determine nefness of divisors on the resulting varieties and to identify when the contractions are K_X-extremal.
• Apply the framework to products of del Pezzo surfaces to produce weak Fano fourfolds and analyze when they are Fano, weak Fano, or of Fano type.

提出的方法

• Introduce a general setting: blow up X'' along A'' and then along the strict transform of B'' to obtain X.
• Describe the relative nef cone and the cone of curves: Nef(X/X'') = R^+[-E,-E-F], NE(X/X'') = R^+[e,f].
• Characterize the contraction psi: X -> X0 as the contraction of the extremal ray R^+[e] and identify X0 as the blowup of X'' along A'' ∪ B''.
• Establish nefness criteria for divisors of the form pullback(H'') - E and pullback(H'') - E - F under suitable stratified nef assumptions (Theorem C).
• Specialize to the product X'' = X1 × X2 of two varieties and derive composite nefness results for divisors like H1 + H2 - E (Propositions 2.4, 2.5, 2.6).
• Apply the construction to products of del Pezzo surfaces to obtain explicit fourfolds with a KX-extremal small contraction and classify them as Fano, weak Fano, or of Fano type (Theorem E).

实验结果

研究问题

• RQ1 Can one construct smooth varieties with K_X-extremal small contractions starting from arbitrary smooth projective varieties?
• RQ2 Under what center configurations and codimension inequalities does the two-step blowup yield a small contraction and how is X0 described?
• RQ3 When are the divisors arising from the two successive blowups nef, and how can they be used to control the birational geometry?
• RQ4 To what extent can these constructions produce weak Fano fourfolds, and how do the resulting properties (Fano, weak Fano, Fano type) depend on the chosen centers and ambient factors?
• RQ5 What new examples of fourfolds with large Picard number can be obtained from products of del Pezzo surfaces exhibiting small contractions?

主要发现

• There exists a birational contraction ψ: X → X0 over X'' with ρ(X/X'') = 1, and X0 is the blowup of X'' along A'' ∪ B'' (Theorem A).
• ψ is a small contraction if and only if A'' is not contained in B'', and ψ is K_X-extremal if and only if a > b (where a,b are codimensions of A'' and B'' in X'').
• The relative nef cone and cone of curves of X over X'' are explicitly given as Ner(X/X'') = R^+[-E,-E-F] and NE(X/X'') = R^+[e,f], with intersection data guiding the contraction.
• Sufficient nefness criteria for divisors on X, such as pullback(H'') - E and pullback(H'') - E - F, are provided under chains of nef restrictions along stratifications, enabling global nefness conclusions (Theorem C).
• When X'' = X1 × X2 and centers are products A'' = A1 × A2, B'' = B1 × B2, the paper proves nefness for divisors like H1 + H2 - E under suitable hypotheses (Lemmas 2.5, 2.6, Proposition 2.4).
• Application to products of del Pezzo surfaces yields explicit fourfolds with K_X-extremal small contractions, and Theorem E gives criteria: X is of Fano type iff r2 ∈ {0,1}, X is weak Fano iff r2 ∈ {0,1} (and not Fano if r1>0 or r2>0), and X is Fano precisely when r1 = r2 = 0; in particular, certain parameter choices yield smooth weak Fano fourfolds with Picard number 8 admitting small contractions.
• The construction extends previous work by Kawamata and Tsukioka, allowing more flexible codimension centers and non-disjoint, higher-dimensional intersections, and it furnishes new non-product examples with large Picard number beyond known cases.