論文の公開元へStudies of compactifications of affine homology 3-cells into del Pezzo fibrations
概要
This thesis concerns complex projective compactifications of smooth affine 3-folds with the same homology rings as that of the affine 3-space.
For an affine variety U, the pair (X, D) of a smooth proper variety X and its reduced effective divisor D is called a compactification of U when the complement X\D is algebraically isomorphic to U. F. Hirzebruch raised the problem to classify all the compactifications (X, D) of the affine n-space A n with second Betti number B2(X) = 1 in his problem list [Hir54]. Here we call this problem the Hirzebruch problem. This problem is trivial when n = 1 because the projective line P 1 is the unique rational smooth proper curve, and when n = 2, it was solved by R. Remmert and T. Van de Ven [RvdV60]. Also by the contribution of M. Furushima, N. Nakayama, Th. Peternell, Y. Prokhorov and M. Schneider [Fur86, Fur90,Fur93a,Fur93b,FN89a,FN89b,Pet89,PS88,Pro91], this problem was solved in the projective case when n = 3. We note that the ambient space X is a Fano variety in the projective case since it is rational with B2 = 1. There is also a generalization of the Hirzebruch problem with B2 ≥ 2, which is studied by several authors [Mor73,MS90,Kis05,Nag18].
In this thesis, we will study three problems which originated from the Hirzebruch problem. It is worth pointing out that up to the present all of them have been investigated only when the dimension is at most 2 or when the ambient spaces are Fano. For this reason, we will discuss three problems when ambient spaces are del Pezzo fibrations, which form a building block of the minimal model program for 3-folds as well as Fano 3-folds.
The first one concerns characterizations of A n . For n ∈ Z>0, an affine homology n-cell is a smooth affine variety of dimension n with the same homology ring as that of A n . By several authors [Ram71, KR97, tDP90], it is known that there are many affine homology n-cells not isomorphic to A n . One of natural questions about affine homology n-cells is how to characterize A n among them. For this question, Furushima [Fur00] pointed out that A 3 can be characterized as the affine homology 3-cell which is compactified into a smooth Fano 3-fold with B2 = 1. Chapter 2 of this thesis gives another characterization of A 3 via compactifications into quadric fibrations, i.e., del Pezzo fibrations of degree 8.
The second problem is a construction of standard maps preserving A n from compactifications of A n to a standard one. In [Mor73], S. Mori introduced three kinds of explicit birational transformations between Hirzebruch surfaces, and showed that any compactifications of A 2 into Hirzebruch surfaces are constructed from the standard compactification (P 2 , P 1 ) of A 2 with finite composition of these birational transformations. Chapter 2 of this thesis deals a construction of standard maps for a certain family of compactifications of A 3 into quadric fibrations, where the standard compactification is the pair (P 3 , P 2 ).
The third problem is the Hirzebruch problem for the affine n-space Gn a equipped with the additive group structure. A Gn a -variety is defined to be a variety with a Gn a -action whose dense orbit is isomorphic to Gn a . The study of smooth projective Gn a -varieties is started by B. Hassett and Y. Tschinkel [HT12], and they classified smooth projective Gn a -varieties with B2 = 1 when n ≤ 3, on which situation Gn a -varieties are Fano. After that, smooth Fano Gn a - varieties are studied by several authors [HM18,FM19]. Chapter 3 discusses the existences of G3 a -structures, i.e., G3 a -actions which give structures of G3 a -varieties, on del Pezzo fibrations.
This thesis consists of three chapters.
Chapter 1 is the preliminary chapter; we recall definitions and basic properties of del Pezzo fibrations and certain elementary links, which we will use throughout this thesis.
Chapter 2 deals with compactifications of affine homology 3-cells into quadric fibrations such that the boundary divisors contain fibers. In this chapter, we show that all such affine homology 3-cells are isomorphic to A 3 , and give explicit birational maps from these compactifications to P 3 preserving A 3 using the technique of elementary links.
Chapter 3 deals with G3 a -varieties with del Pezzo fibration structures. In this chapter, we show that del Pezzo fibrations admit G3 a -structures if and only if they are P 2 -bundles.
Chapter 2 and 3 are based on papers [Nag19b] and [Nag19a] respectively.
