具有莱夫谢茨缺陷2的法诺簇的几何
On the geometry of Fano varieties with Lefschetz defect 2
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中文总结 AI 辅助
本文研究莱夫谢茨缺陷为2的光滑复法诺簇的几何，通过翻转序列得到其圆锥丛结构，分析圆锥丛的光滑性或德·佩佐纤维化性质，最终构造出表现该行为的法诺四维簇族。
中文摘要 AI 辅助
设X为光滑复法诺簇，δ(X)为其莱夫谢茨缺陷。已知若δ(X)≥4，则X同构于曲面S与T的乘积S×T；若δ(X)=3，则X具有带有特定结构的德·佩佐纤维化。本文研究δ(X)=2的情形。首个结果是存在翻转序列X→X'，使得X'光滑且具有圆锥丛结构X'→Y，满足ρ(X)-ρ(Y)=2。该圆锥丛分解为X'→X''→Y，其中X'→X''是光滑爆破，f:X''→Y仍为圆锥丛。随后证明要么f是光滑的，要么存在德·佩佐纤维化g:X→T，满足ρ(X)-ρ(T)=3且T光滑，并描述了相对锥NE(g)。最后，针对其中一种可能的一般纤维，给出德·佩佐纤维化g的结构定理，证明T是法诺簇，且X可由T及其上的合适除子类重构。作为应用，构造了表现此行为的法诺四维簇族。
英文摘要
Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if delta(X) is at least 4, then X is isomorphic to a product SxT where S is a surface, whereas if delta(X)=3, X admits a del Pezzo fibration with a prescribed structure. In this paper, we study the case where delta(X)=2. Our first result is that there exists a sequence of flips X-->X' such that X' is smooth and admits a conic bundle structure X'->Y, with rho(X)-rho(Y)=2. The conic bundle factors as X'->X''->Y, where X'->X'' is a smooth blow-up, and f: X''->Y is again a conic bundle. Then we show that either f is smooth, or there exists a del Pezzo fibration g: X->T with rho(X)-rho(T)=3 and T smooth, admitting two possible generic fibers, and we describe the relative cone NE(g). Finally for one of the possible generic fibers we give a structure theorem for the del Pezzo fibration g; we show that T is Fano and that X can be reconstructed from T and a suitable divisor class on T. As an application, we construct a family of Fano 4-folds X exhibiting this behaviour.
