Sarkisov Links with centres space curves on smooth cubic surfaces

Sarkisov links for index 1 Fano 3-folds in codimension 4

We classify Sarkisov links from index 1 Fano 3-folds anticanonically embedded in codimension 4 that start from so-called Type I Tom centres. We apply this to compute the Picard rank of many such Fano 3-folds.

从对数卡拉比-丘几何视角研究非奇异平面三次曲线的分解群

On the decomposition group of a nonsingular plane cubic by a log Calabi-Yau geometrical perspective

本文从对数卡拉比-丘几何视角研究非奇异平面三次曲线的分解群，证明二维Sarkisov纲领算法对该群元素自动保体积，并给出三维情形下Pan定理推广的反例。

Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

We characterise smooth curves in P^3 whose blow-up produces a threefold with anticanonical divisor big and nef. These are curves C of degree d and genus g lying on a smooth quartic, such that (i) $4d-30 \le g\le 14$ or $(g,d) = (19,12)$, (ii) there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to C. This generalises the classical similar situation for the blow-up of points in P^2. We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.

Explicit Birational Geometry of Fano threefold complete intersections

We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family $X$ has Fano index at least 2 and is minimally embedded in a weighted projective space in codimension 2, we determine which cyclic quotient singularity is a maximal centre. If a cyclic quotient singularity is a maximal centre, we construct a Sarkisov link to a non-isomorphic Mori fibre space or a birational involution. This allows, in particular, the construction of new examples of Fano 3-folds of co…

On birational maps from cubic threefolds

We characterise smooth curves in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree $d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a 3-secant line in the cubic threefold. Among the list of ten possible such types $(g,d)$, two were previously left as open numerical possibilities, namely $(g,d) = (0,5)$ and $(2,6)$. Using the Sarkisov link associated with a curve of type $(2,6)$, we are able to produce the first example of a pseudo-automorphism with dynamical degree greater than $1$ on a smooth threefold wi…

球体堆积中的互补体

Complementary bodies in sphere packing

本文研究立方密堆积中空隙形成的球截多面体互补体，推导其表面积与体积比，并证明截断四面体和八面体如何铺砌球间空隙，深化了对经典堆积问题及其几何互补体的理解。