Singularities of linear systems and 3-fold birational geometry

Weighted Fano threefold hypersurfaces

“…Let V ⊂ P(1, a 4 , a 5 ) be the open subset given by x 1 = 0. Then V ∼ = C 2 and the affine curve V ∩ C is either a cubic curve when ⌊d/a 4 ⌋ 3 or a double cover of C ramified at most four points when ⌊d/a 4 ⌋ 4 and 2a 5 ≤ d < 2a 5 + a 4 .…”

“…Let D be a general surface of in the linear system | − 5K X | and S be the unique surface of the linear system | − K X |. Then D is cut on X by the equation λx 5 1 + δx 1 x 2 + µx 3 = 0, where (λ : δ : µ) ∈ P 2 , and S is cut by the equation x 1 = 0. Moreover, the base locus of the linear system | − 5K X | consists of the single irreducible curve C that is cut on the hypersurface X by the equations x 1 = x 3 = 0.…”

“…Therefore, it follows from [6] that CS(X, λM) = {P 2 , P 3 }. 5 Fix a very ample linear system H on X. Let φ : X X be a birational automorphism such that φ −1 (H) ⊂ | − rKX |.…”

“…where ψ is the natural projection, α 4 is the weighted blow up at the singular point P 4 with weights (1, 1, 5), α 5 is the weighted blow up at the point P 5 with weights (1, 2, 5), β 4 is the weighted blow up with weights (1,1,5) at the point α −1 5 (P 4 ), β 5 is the weighted blow up with weights (1,2,5) at the point α −1 4 (P 5 ), and η is an elliptic fibration. It follows from [6] …”

Бирациональная Геометрия Двойных Накрытий Фано

“…Понятие бирациональной (сверх)жесткости является об-щепринятым, однако разные авторы используют разные определения (см. [2], [8]- [10]). По-видимому, наиболее точным является такое определение: про-ективное рационально связное многообразие X бирационально жесткое, если существует модель X , бирационально эквивалентная X , удовлетворяющая условию п.…”