A simple proof of Grothendieck’s theorem on the parafactoriality of local rings

Birational geometry of Fano hypersurfaces of index two

“…Indeed, it follows from (R4) that the section of V by a generic hyperplane, containing the subspace P, in non-singular, whereas it is well known that any hyperplane section of a non-singular hypersurface in a projective space can have at most isolated singularities. As the fibres of π are isomorphic to the sections of V by hyperplanes, containing the subspace P, we conclude that V + has at most finitely many isolated singular points, hence V + is factorial by Grothendieck's theorem [1]. Those isolated singularities are double points as they are double points on the corresponding fibres of π by the condition (R1).…”

“…Proof These claims are obvious by the regularity condition (R4) and the well known factoriality of an isolated hypersurface singularity in the dimension 4 and higher, see [1]. (Note that the smoothness of V is contained in the regularity condition (R1).)…”

“…Therefore, its minimum on that interval is equal to min{ f (2), f (N − 1)}, and elementary calculations show that both numbers are higher than 1 2 N (N + 1) + 2. Proof In the notations of the condition (R1) it is sufficient to consider the worst case c = 3.…”

“…By the inequality (6), the integer l * satisfies the inequality l * > 4 3 m. Now, repeating the arguments in the beginning of Sec. 3.3 in [22] word for word (using the regularity condition (R3) instead of the condition (R2.2) in [22]), we exclude the case (1). Now let us consider the hardest case (2).…”

Бирационально Жесткие Многообразия. II. Расслоения Фано

“…Эти утверждения непосредственно вытекают из опреде-ления раздутия ϕ, общности многообразия V и следующего хорошо известного факта: изолированная гиперповерхностная особенность многообразия размер-ности 4 факториальна (см. [49], [22]). …”