Miniworkshop
Explicit Birational and Affine Geometry
in Saitama
November 1617 2018, Department of Mathematics, Saitama University
Program

On a question of Charlie Stibitz and Ziquan Zhuang
Ivan CHELTSOV (University of Edinburgh)
Abstract
In https://arxiv.org/abs/1802.08381, Charlie Stibitz and Ziquan Zhuang posed a nice question: is it true that
alphainvariants of birationally superrigid Fano varieties are greater than $1/2$ ? In my talk, I will explain that the answer to this question is probably No.
This is a joint work with Ziquan Zhuang (Princeton)

Cylinders in forms of the quintic del Pezzo threefold
Adrien DUBOULOZ (University of Burgundy)
Abstract
Completions of the affine space $\mathbb{A}_{\mathbb{C}}^{3}$ into
the quintic del Pezzo threefold $V_{5}$, the smooth Fano threefold
of index two and degree five, have been studied and classified by
Furushima and Nakayama in the late eighties. In this talk, we consider
the problem of existence of $\mathbb{A}^{n}$cylinders, $n=1,2,3$,
in forms of $V_{5}$ over nonclosed fields $k$ of characteristic
zero. The upshot is that such a form $X$ always contains an $\mathbb{A}_{k}^{2}$cylinder
but contains $\mathbb{A}_{k}^{3}$ as a Zariski open subset if and
only if $X$ contains a line defined over $k$ with nontrivial normal
bundle. I will explain how to explicitly construct these cylinders
via Sarkisov links performed from certain locally trivial $\mathbb{P}^{1}$bundles
over $\mathbb{P}^{2}$ associated to standard birational quadratic
involutions of $\mathbb{P}^{2}$. Joint work with Takashi Kishimoto.

Twisted Cylinders in Mori Fiber Spaces
Takashi KISHIMOTO (Saitama University)
Abstract
Cylinders found in projective varieties provide us an effective tool to produce various actions of unipotent groups on affine cones over them.
For this purpose, it is, in some sense, essential to find cylinders contained in Mori Fiber Space (MFS). Letting $\pi: V \to W$ be MFS, certainly it seems
to be reasonable to try finding cylinders which respect the structure of $\pi$, however such cylinders can not always exist, e.g., for the case in which
general fibers of $\pi$ are birationally rigid. In this talk, we will show: for any of $95$ pairs of positive integers $1 \leqq a_1 \leqq a_2 \leqq a_3 \leqq a_4$
found in the list of Fletcher and for any homogeneous polynomial $h \in \mathbb{C}[x_1, \cdots , x_4]$ of degree $d =\sum_{i=1}^4 a_i$ defining a
quasismooth surface in $\mathbb{P} (a_1, \cdots , a_4)\cong {\rm Proj}(\mathbb{C}[x_1, \cdots , x_4])$, we shall construct a fourdimensional
MFS $\pi_h : V_h \to \mathbb{P}_{\mathbb{C}}^1$ such that general fibers of $\pi_h$ are isomorphic to quasismooth weighted hypersurfaces
$\{ h \lambda x_0^d =0 \}$ of degree $d$ in $\mathbb{P} (1,a_1, \cdots , a_4)\cong {\rm Proj} (\mathbb{C} [x_0, x_1, \cdots , x_4])$, hence birationally
rigid $\mathbb{Q}$Fano threefolds, nevertheless the total space $V_h$ itself contains the affine $4$space $\mathbb{A}_{\mathbb{C}}^4$.
This is a joint work with Adrien Dubouloz and Karol Palka.

On compactifications of contractible affine threefolds into del Pezzo fibrations
Masaru NAGAOKA (University of Tokyo)
Abstract
By the contribution of M. Furushima, N. Nakayama, Th. Peternell and M. Schneider, it is completed to classify all projective compactifications of the
affine $3$space $\mathbb{A}^3$ with Picard number one. After that, T. Kishimoto observed that their arguments make use of only the contractibility of
$\mathbb{A}^3$ and that the compactifications are Fano manifolds. In this talk, I will consider compactifications of contractible affine
threefolds into another special manifolds, i.e. del Pezzo fibrations. Mainly I will introduce certain type of such compactification,
which seems to have connection with vertical cylinder, and give you examples of such certain compactification.

Sextic elliptic curves on prime Fano 3folds of genus 9 or 10
Takeru FUKUOKA (University of Tokyo)
Abstract
In this talk, we will discuss about sextic elliptic curves on prime Fano 3folds of genus 9 or 10. By Mukai's classification, a prime Fano 3folds of genus 9 (resp. 10) is always a linear section of the 5dimensional homogeneous manifold of
type G2 (resp. the Lagrangian Grassmannian of a symplectic 6dimensional
vector space). Such homogeneous manifold naturally has a certain vector bundle of rank
2 (resp. rank 3) and the zero scheme of a general global section is isomorphic to a sextic del Pezzo 3fold.
By cutting them, we will obtain an elliptic curve of degree 6 on a prime
Fano 3fold of genus 9 (resp. 10). The main result of this talk gives an explicit description of the 2ray
game of the blowingup of such prime Fano 3folds along these elliptic curves.

Birational rigidity of del Pezzo fibrations
Takuzo OKADA (Saga University)
Abstract
The aim of my talk is to explain subtlety in (various) definitions of "birational rigidity" for del Pezzo fibrations.
We have two closely related notions: birational rigidity and birational rigidity over the base. Sometimes both of these two notions are simply called
birationally rigid and this leads to a confusion. I will explain their difference by giving concrete examples and then explain that we need to
be careful when stating the famous conjecture, $K$conjecture, on birational rigidity of del Pezzo fibrations.
Tentative Schedule
Friday 
Saturday 

OKADA 10:3011:30 
CHELTSOV 11:5012:50

Lunch Break 
NAGAOKA 15:0016:00 
DUBOULOZ 14:3015:30 
FUKUOKA 16:2017:20 
KISHIMOTO 15:5016:50 
Practical Informations
Contacts
If you want to participate or for more information, please contact:
Takashi KISHIMOTO
Venue
The workshop will take place at Kisosuriensyushitsu; Falculty of Sciences, 3rd floor of the building no. 14 on
the campus map.
Poster
Organizers and supports
Takashi Kishimoto, Ivan Cheltsov and Adrien Dubouloz