Fano 3-fold hypersurfaces

Weighted Fano threefold hypersurfaces

“…• This theorem with the results of [6] can be considered as a 3-fold analogue of Yu. Manin's results on smooth del Pezzo surfaces of degree ≤ 3.…”

“…Let C be a general fiber of the projection X P(1, a 2 , a 3 ). Then C is not a rational curve by [6] but C is a hypersurface of degree…”

“…Then there is a linear system M without fixed components on the hypersurface X such that the set CS(X, λM) contains at least two subvariety of the hypersurface X, where λ is a positive rational number such that the divisor −(K X + λM) is ample. Therefore, it follows from [6] that CS(X, λM) = {P 2 , P 3 }. 5 Fix a very ample linear system H on X.…”

“…Iano-Fletcher, J. Johnson, J. Kollár, and M. Reid (see [9] and [11]) and which were studied quite extensively in [6] and [18].…”

“…Also, when X has a unique birational involution, the group Γ X has the presentation F 1 that is isomorphic to Z/2Z. Because the number of generators of Γ X is completely determined in [6], in order to describe the group Γ X , it is enough to find their relations for ℓ ≥ 2. We prove the following result: Theorem 1.1.…”