In this paper, we study Du Val del Pezzo surfaces over an algebraically closed field of positive characteristic. Such a surface X may satisfy the condition (NB): all the anti-canonical divisors are singular, (ND): there are no Du Val del Pezzo surfaces over the field of complex numbers with the same Dynkin type and Picard rank, and (NL): the pair (Y, E) does not lift to the ring of Witt vectors, where Y is the minimal resolution and E is its reduced exceptional divisor. On one hand, we show that (ND) ⇒ (NL) ⇒ (NB). On the other hand, for each condition above, we determine all the Du Val del Pezzo surfaces with the given condition. We also give the list of the automorphism groups of Du Val del Pezzo surfaces satisfying one of them. Furthermore, following Ye's method of classification of Du Val del Pezzo surfaces in characteristic zero, we also classify Du Val del Pezzo surfaces of the Picard rank one in characteristic two or three. Finally, we show that if Du Val del Pezzo surfaces are Frobenius split, then they satisfy none of the conditions above.

Kishimoto raised the problem to classify all compactifications of contractible affine [Formula: see text]-folds into smooth Fano [Formula: see text]-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this paper, we show that there are [Formula: see text] deformation equivalence classes of smooth Fano [Formula: see text]-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes.

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T. Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this article, we show that there are 14 deformation equivalence classes of smooth Fano 3-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes. Contents 1. Introduction. 2. Preliminaries 2.1. Gorenstein del Pezzo surfaces 2.2. Topologies of varieties 3. Boundary divisors 4. Construction of examples of type (A) 4.1. Affine modifications 4.2. The types (A1) and (A2) 4.3. The type (A3) 4.4. The type (A4) 4.5. Another example of type (A3) 5. Construction of examples of type (B) 5.1. The characterization of A 3 5.2. The type (B3) 5.3. The types ( B1) and (B2) 6. Exclusion of imprimitive Fano 3-folds 6.1. The image of contraction ϕ 1 : V → W 1 6.2. On the conditions in Lemma 4.3 6.3. The case W 1 = P 3 6.4. The case W 1 = Q 3 6.5. The case W 1 = V 5 7. Exclusion of primitive Fano 3-folds References

In this paper we deal with compactifications of affine homology 3-cells into quadric fibrations such that the boundary divisors contain fibers. We show that all such affine homology 3-cells are isomorphic to the affine 3-space A 3 . Moreover, we show that all such compactifications can be connected by explicit elementary links preserving A 3 to the projective 3-space P 3 . Contents 1. Introduction. 1 2. Elementary links 6 2.1. Elementary links between blow-ups 7 2.2. Elementary links between P 2 -bundles 7 2.3. Elementary links from quadric fibrations to P 2 -bundles 8 2.4. Elementary links between quadric fibrations 10 3. Topological invariants of the ambient space 11 4. Proof of Theorem 1.3. 11 5. Examples 14 6. Compactifications of affine homology 3-cells compatible with P 2 -bundles 16 7. Proof of Theorem 1.8 18 7.1. Singularities of D h and D f | D h 18 7.2. The case of singular D f | D h 21 7.3. The case of smooth D f | D h 22 Acknowledgement 23 References 24