K-stable Fano threefolds of rank 2 and degree 30

Abstract

We find all K-stable smooth Fano threefolds in the family No. 2.22.

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

“…In the book [Ara+23], it is completely determined whether the general member of each of the 105 irreducible families of smooth Fano 3-folds admits a KE metric or not. Very recently, there have been a lot of works aiming to drop the generality assumption in the above result and to classify K-(poly)stable smooth Fano 3-folds in each family completely (see [Liu23], [CP22], [CFKO22], [CFKP23], [BL22], [Den22], [CDF22], [Mal23]).…”

K-stable smooth Fano threefolds of Picard rank two

“…If 𝑢 ∈ 𝐼 𝑛,4 and 𝑣 ∈ 0, 31+58𝑛+28𝑛 2 −𝑢 (30+58𝑛+28𝑛 2 ) 182𝑛 + 84𝑛 2 − 𝑢 100 + 182𝑛 + 84𝑛 2 − 𝑣 36 + 56𝑛 + 21𝑛 2 36 + 56𝑛 + 21𝑛 2 e 1 + + 17 + 32𝑛 + 14𝑛 2 − 𝑢 20 + 34𝑛 + 14𝑛 2 36 + 56𝑛 + 21𝑛 2 𝐵 𝑛,4 + 𝑢 48 + 98𝑛 + 49𝑛 2 − 39 − 91𝑛 − 49𝑛 2 36 + 56𝑛 + 21𝑛 2 𝐵 𝑛+1,1and 𝑁 (𝑢, 𝑣) = 0. The same holds when 𝑢 ∈ 𝐼 𝑛,4 and 𝑣 ∈ 0, 44+70𝑛+28𝑛 2 −𝑢 (43+70𝑛+28𝑛 2 )If 𝑢 ∈ 𝐼 𝑛,4 and 𝑣 ∈ 31+58𝑛+28𝑛 2 −𝑢 (30+58𝑛+28𝑛 2 ) 182𝑛 + 84𝑛 2 − 𝑢(100 + 182𝑛 + 84𝑛 2 ) − 𝑣(36 + 56𝑛 + 21𝑛 2 ) 36 + 56𝑛 + 21𝑛 2 e 1 + + (1 + 𝑛)(11 + 7𝑛) 103 + 182𝑛 + 84𝑛 2 − 𝑢(100 + 182𝑛 + 84𝑛 2 ) − 𝑣(36 + 56𝑛 + 21𝑛 2 )Similarly, we get𝑀 𝑛,1 = (1 + 7𝑛 + 7𝑛 2 ) 𝐴 𝑛,128(1 + 𝑛)(1 + 3𝑛)4 (2 + 7𝑛)4 (3 + 7𝑛)4 (4 + 7𝑛)4 ,where𝐴 𝑛,1 = 480574 + 12906866𝑛 + 157271760𝑛 2 + 1149521334𝑛 3 + 5612285145𝑛 4 + + 19278934535𝑛 5 + 47770884833𝑛 6 + 86016481159𝑛 7 + 111679016743𝑛 8 + + 101939513907𝑛 9 + 62077730148𝑛 10 + 22635902898𝑛 11 + 3735591048𝑛 12 . 14𝑛 + 7𝑛 2 ) 𝐴 𝑛,2 224(1 + 𝑛)(1 + 2𝑛) 3 (2 + 7𝑛) 4 (3 + 7𝑛) 4 (4 + 7𝑛) 4 ,and𝑀 𝑛,2 = 11780 + 111142𝑛 + 430951𝑛 2 + 875637𝑛 3 + 978656𝑛 4 + 566832𝑛 5 + 131712𝑛 6 224(1 + 2𝑛) 3 (3 + 7𝑛) 4 (6 + 14𝑛 + 7𝑛 2 ) ,where𝐴𝑛,2 = 1561176 + 35176776𝑛 + 356105548𝑛 2 + 2137950448𝑛 3 + + 8458603286𝑛 4 + 23158717414𝑛 5 + 44778314889𝑛 6 + 61151030584𝑛 7 + + 57807289939𝑛 8 + 36026947376𝑛 9 + 13321631568𝑛 10 + 2213683584𝑛 11 .…”

“…Therefore, there exists a prime divisor E over X such that 𝐶 𝑋 (E) is a curve, 𝑃 ∈ 𝐶 𝑋 (E), and 𝐴 𝑋 (E) <4 3 𝑆 𝑋 (E). Set 𝑍 = 𝐶 𝑋 (E).…”

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

“…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]…”