Let
be an imperfect field.
Let
be a regular variety over
and
set
to be the normalization of
.
In this paper, we show that
for some effective divisor
on
.
We obtain the following three applications.
...First, we show that a
-trivial fiber space with non-normal fibers is uniruled.
Second, we prove that general fibers of Mori fiber spaces are rationally chain connected.
Third, we obtain a weakening of the cone theorem for surfaces and threefolds defined over an imperfect field.
For a regular del Pezzo surface
X
, we prove that
|
-
12
K
X
|
is very ample. Furthermore, we also give an explicit upper bound for the volume
K
X
2
which depends only on
k
:
k
p
for the base field
...k
. As a consequence, we obtain the boundedness of geometrically integral regular del Pezzo surfaces.
Given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmüller ...lift of an ample invertible sheaf. We also give a generalisation of this vanishing theorem to one of Kawamata–Viehweg type.
In this note, we study base point freeness up to taking p‐power, which we will call p‐power freeness. We first establish some criteria for p‐power freeness as analogues of criteria for ...semi‐ampleness. We then apply these results to three‐dimensional birational geometry.
We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show ...the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.
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