Seshadri constants and K-stability of Fano manifolds

K-stable smooth Fano threefolds of Picard rank two

“…Therefore, integrating ( 𝑃(𝑢, 𝑣) • 𝐸) 2 , we obtain 𝑆(𝑊 𝑌 , 𝑆 •,•,• ; 𝑂) = 13 24 + 𝐹 𝑂 . If 𝑂 ∉ 𝐶 ∪ 𝑅, then 𝐹 𝑂 = 0.…”

“…Then 𝜋(𝑆) ∩ Q is cut out on the surface 𝜋(𝑆) by 𝑐 1 𝑥 + 𝑐 2 𝑦 + 𝑐 3 𝑧 + 𝑐 4 𝑡 = 0, where 𝑐 1 , 𝑐 2 , 𝑐 3 , 𝑐 4 are general numbers. The affine part of the surface 𝜋(𝑆) is isomorphic to the hypersurface in C 3 given by 𝑎𝑔 2 2 (𝑥, 𝑦, 𝑧) + 𝑏𝑥𝑔 2 (𝑥, 𝑦, 𝑧) + 𝑓 2 (𝑥, 𝑦, 𝑧) = 0, and the affine part of the curve 𝜋(𝑆) ∩ Q is cut out by 𝑐 1 𝑥 + 𝑐 2 𝑦 + 𝑐 3 𝑧 + 𝑐 4 𝑔 2 (𝑥, 𝑦, 𝑧) = 0. If P is a singular point of the surface S of type D 4 or D 5 , then 𝑆 ∩ 𝑇 has an ordinary cusp at the point P, which easily implies that the intersection 𝑆 ∩ 𝑇 is reduced and irreducible.…”

On K$K$‐stability of P3$\mathbb {P}^3$ blown up along the disjoint union of a twisted cubic curve and a line

“…The intersections are given by: 2 , which implies that −𝐾 𝑋 − 𝑢𝑆 is not pseudoeffective for 𝑢 > 3∕2. Let 𝑃(𝑢) = 𝑃(−𝐾 𝑋 − 𝑢𝑆) be a positive part of Zariski decomposition and 𝑁(𝑢) = 𝑁(−𝐾 𝑋 − 𝑢𝑆) be a negative part of Zariski decomposition.…”

“…Let us seek for a contradiction. Let us identify 𝜋() = ℙ 1 × ℙ 1 such that 𝐶 is a curve in 𝜋() of degree (1,2). Then, 𝜋 induces a birational morphism 𝜑 ∶  → ℙ 1 × ℙ 1 that is a blowup of two intersection points 𝜋() ∩ 𝐿 = {𝐴 1 , 𝐴 2 }, which are not contained in the curve 𝐶.…”

“…This problem has already been solved for a number of families of Fano threefolds (see [2–7, 13, 16, 17, 24]). In this article, we will solve the Calabi problem completely for Family 3.12, by using recent advances in estimating delta invariants [1], combined with previous partial results on this family presented in [3, §5.18]…”