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We study dynamics of solutions in the initial value space of the sixth Painlevé equation as the independent variable approaches zero. Our main results describe the repeller set, show that the number of poles and zeroes of general solutions is unbounded and that the complex limit set of each solution exists and is compact and connected.
We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .
For each central essential hyperplane arrangement
$\mathcal{A}$
over an algebraically closed field, let
$Z_\mathcal{A}^{\hat\mu}(T)$
denote the Denef–Loeser motivic zeta function of
$\mathcal{A}$
. We prove a formula expressing
$Z_\mathcal{A}^{\hat\mu}(T)$
in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which
$Z_{\mathcal{A}}^{\hat\mu}(T)$
fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, we use this formula to show that the map taking each complex arrangement
$\mathcal{A}$
to the Hodge–Deligne specialization of
$Z_{\mathcal{A}}^{\hat\mu}(T)$
is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of
$\mathcal{A}$
in terms of the characteristic polynomials of related arrangements.
We prove some
$\ell $
-independence results on local constancy of étale cohomology of rigid analytic varieties. As a result, we show that a closed subscheme of a proper scheme over an algebraically closed complete non-archimedean field has a small open neighbourhood in the analytic topology such that, for every prime number
$\ell $
different from the residue characteristic, the closed subscheme and the open neighbourhood have the same étale cohomology with
${\mathbb Z}/\ell {\mathbb Z}$
-coefficients. The existence of such an open neighbourhood for each
$\ell $
was proved by Huber. A key ingredient in the proof is a uniform refinement of a theorem of Orgogozo on the compatibility of the nearby cycles over general bases with base change.
Let
$(X\ni x,B)$
be an lc surface germ. If
$X\ni x$
is klt, we show that there exists a divisor computing the minimal log discrepancy of
$(X\ni x,B)$
that is a Kollár component of
$X\ni x$
. If
$B\not=0$
or
$X\ni x$
is not Du Val, we show that any divisor computing the minimal log discrepancy of
$(X\ni x,B)$
is a potential lc place of
$X\ni x$
. This extends a result of Blum and Kawakita who independently showed that any divisor computing the minimal log discrepancy on a smooth surface is a potential lc place.
Let V be a smooth quasi-projective complex surface such that the first three logarithmic plurigenera
$\overline P_1(V)$
,
$\overline P_2(V)$
and
$\overline P_3(V)$
are equal to 1 and the logarithmic irregularity
$\overline q(V)$
is equal to
$2$
. We prove that the quasi-Albanese morphism
$a_V\colon V\to A(V)$
is birational and there exists a finite set S such that
$a_V$
is proper over
$A(V)\setminus S$
, thus giving a sharp effective version of a classical result of Iitaka [12].
We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups:
${\mathfrak{A}}_5$
,
${\text{PSL}}_2(\textbf{F}_7)$
,
${\mathfrak{A}}_6$
,
${\text{SL}}_2(\textbf{F}_8)$
,
${\mathfrak{A}}_7$
,
${\text{PSp}}_4(\textbf{F}_3)$
,
${\text{SL}}_2(\textbf{F}_{7})$
,
$2.{\mathfrak{A}}_5$
,
$2.{\mathfrak{A}}_6$
,
$3.{\mathfrak{A}}_6$
or
$6.{\mathfrak{A}}_6$
. All of these groups with a possible exception of
$2.{\mathfrak{A}}_6$
and
$6.{\mathfrak{A}}_6$
indeed act on some rationally connected threefolds.
In this note, we prove the semiampleness conjecture for Kawamata log terminal Calabi–Yau (CY) surface pairs over an excellent base ring. As applications, we deduce that generalized abundance and Serrano’s conjecture hold for surfaces. Finally, we study the semiampleness conjecture for CY threefolds over a mixed characteristic DVR.
The pentagram map, introduced by Schwartz [The pentagram map. Exp. Math.1(1) (1992), 71–81], is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev’s proof of complex integrability [F. Soloviev. Integrability of the pentagram map. Duke Math. J.162(15) (2013), 2815–2853]. In the course of the proof, we construct the moduli space of twisted n-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.
A famous problem in birational geometry is to determine when the birational automorphism group of a Fano variety is finite. The Noether–Fano method has been the main approach to this problem. The purpose of this paper is to give a new approach to the problem by showing that in every positive characteristic, there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphism group. To do this, we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Kollár produces on $p$-cyclic covers in characteristic $p > 0$.
We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups.
We study lc pairs polarized by a nef and log big divisor. After proving the minimal model theory for projective lc pairs polarized by a nef and log big divisor, we prove the effectivity of the Iitaka fibrations and some boundedness results for dlt pairs polarized by a nef and log big divisor.
We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation
$A^2-DB^2=1$
, with
$A,B,D\in \mathbb {C}[t]$
and certain ramified covers
$\mathbb {P}^1\to \mathbb {P}^1$
arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Corvaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomials D that fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann existence theorem associates to the abovementioned covers certain permutation representations: We are able to characterize the representations corresponding to ‘primitive’ solutions of the Pell equation or to powers of solutions of lower degree and give a combinatorial description of these representations when D has degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map.
We prove rationality criteria over nonclosed fields of characteristic
$0$
for five out of six types of geometrically rational Fano threefolds of Picard number
$1$
and geometric Picard number bigger than
$1$
. For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type.
Let $\pi \colon \mathcal {X}\to B$ be a family whose general fibre $X_b$ is a $(d_1,\,\ldots,\,d_a)$-polarization on a general abelian variety, where $1\leq d_i\leq 2$, $i=1,\,\ldots,\,a$ and $a\geq 4$. We show that the fibres are in the same birational class if all the $(m,\,0)$-forms on $X_b$ are liftable to $(m,\,0)$-forms on $\mathcal {X}$, where $m=1$ and $m=a-1$. Actually, we show a general criteria to establish whether the fibres of certain families belong to the same birational class.
We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras which are not of finite type and which arise from a star shaped quiver.
We prove that any nef $b$-divisor class on a projective variety defined over an algebraically closed field of characteristic zero is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef $b$-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint-free curve class is a power of a nef $b$-divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of $b$-divisors which were defined in our previous work.
We study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123–1158). We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non-Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface S which determine a non-Galois triple cover of S. The examples presented are in any admissible Kodaira dimension, and in particular, we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type.
Let
$\Gamma $
be a finite set, and
$X\ni x$
a fixed kawamata log terminal germ. For any lc germ
$(X\ni x,B:=\sum _{i} b_iB_i)$
, such that
$b_i\in \Gamma $
, Nakamura’s conjecture, which is equivalent to the ascending chain condition conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor E over
$X\ni x$
, such that
$a(E,X,B)=\mathrm {mld}(X\ni x,B)$
, and
$a(E,X,0)$
is bounded from above. We extend Nakamura’s conjecture to the setting that
$X\ni x$
is not necessarily fixed and
$\Gamma $
satisfies the descending chain condition, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of
$a(E,X,0)$
for any such E.