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We compactify and regularise the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system has certain unusual properties, including a sequence of points of indeterminacy in
$\mathbb {P}^{1}\!\times \mathbb {P}^{1}$
. These indeterminacy points lie on a singular fibre of the mapping to a corresponding QRT system and provide the existence of a one-parameter family of special solutions.
Let k be a field,
$x_1, \dots , x_n$
be independent variables and let
$L_n = k(x_1, \dots , x_n)$
. The symmetric group
$\operatorname {\Sigma }_n$
acts on
$L_n$
by permuting the variables, and the projective linear group
$\operatorname {PGL}_2$
acts by
$$ \begin{align*} \begin{pmatrix} a & b \\ c & d \end{pmatrix}\, \colon x_i \longmapsto \frac{a x_i + b}{c x_i + d} \end{align*} $$
for each
$i = 1, \dots , n$
. The fixed field
$L_n^{\operatorname {PGL}_2}$
is called “the field of cross-ratios”. Given a subgroup
$S \subset \operatorname {\Sigma }_n$
, H. Tsunogai asked whether
$L_n^S$
rational over
$K_n^S$
. When
$n \geqslant 5,$
the second author has shown that
$L_n^S$
is rational over
$K_n^S$
if and only if S has an orbit of odd order in
$\{ 1, \dots , n \}$
. In this paper, we answer Tsunogai’s question for
$n \leqslant 4$
.
The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic
$D$
-varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose
$\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$
are dominant rational maps from an (possibly nonreduced) irreducible scheme
$Z$
of finite type to an algebraic variety
$X$
, with the property that there are infinitely many hypersurfaces on
$X$
whose scheme-theoretic inverse images under
$\unicode[STIX]{x1D719}_{1}$
and
$\unicode[STIX]{x1D719}_{2}$
agree. Then there is a nonconstant rational function
$g$
on
$X$
such that
$g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$
. In the case where
$Z$
is also reduced, the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou–Hrushovski theorem to generalised algebraic
${\mathcal{D}}$
-varieties and of Cantat’s theorem to self-correspondences.
We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.
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.
For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the Hilbert schemes Hilbn (S) and Hilbn (Sτ) are birational.
We prove the deformation invariance of Kodaira dimension and of certain plurigenera and the existence of canonical models for log surfaces which are smooth over an integral Noetherian scheme
$S$
.
In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over
$\mathbb{F}_{q}$
is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by
$q$
.
Many phenomena in geometry and analysis can be explained via the theory of
$D$
-modules, but this theory explains close to nothing in the non-archimedean case, by the absence of integration by parts. Hence there is a need to look for alternatives. A central example of a notion based on the theory of
$D$
-modules is the notion of holonomic distributions. We study two recent alternatives of this notion in the context of distributions on non-archimedean local fields, namely
$\mathscr{C}^{\text{exp}}$
-class distributions from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] and WF-holonomicity from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We answer a question from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)] by showing that each distribution of the
$\mathscr{C}^{\text{exp}}$
-class is WF-holonomic and thus provides a framework of WF-holonomic distributions, which is stable under taking Fourier transforms. This is interesting because the
$\mathscr{C}^{\text{exp}}$
-class contains many natural distributions, in particular, the distributions studied by Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We show also another stability result of this class, namely, one can regularize distributions without leaving the
$\mathscr{C}^{\text{exp}}$
-class. We strengthen a link from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] between zero loci and smooth loci for functions and distributions of the
$\mathscr{C}^{\text{exp}}$
-class. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory.
We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index
$e$
, then the degree of irrationality of a very general complex Fano hypersurface of index
$e$
and dimension n is bounded from below by a constant times
$\sqrt{n}$
. To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic
$p$
argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.
In this note, using methods introduced by Hacon et al. [‘Boundedness of varieties of log general type’, Proceedings of Symposia in Pure Mathematics, Volume 97 (American Mathematical Society, Providence, RI, 2018) 309–348], we study the accumulation points of volumes of varieties of log general type. First, we show that if the set of boundary coefficients Λ satisfies the descending chain condition (DCC), is closed under limits and contains 1, then the corresponding set of volumes satisfies the DCC and is closed under limits. Then, we consider the case of ε-log canonical varieties, for 0 < ε < 1. In this situation, we prove that if Λ is finite, then the corresponding set of volumes is discrete.
We develop some foundational results in a higher-dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman–Mori cone of curves in terms of the numerical properties of
$K_{{\mathcal{F}}}$
for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program (MMP) for rank 2 foliations on threefolds.
We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.
We provide evidence for this conclusion: given a finite Galois cover
$f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$
of group
$G$
, almost all (in a density sense) realizations of
$G$
over
$\mathbb{Q}$
do not occur as specializations of
$f$
. We show that this holds if the number of branch points of
$f$
is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of
$\mathbb{Q}$
of given group and bounded discriminant. This widely extends a result of Granville on the lack of
$\mathbb{Q}$
-rational points on quadratic twists of hyperelliptic curves over
$\mathbb{Q}$
with large genus, under the abc-conjecture (a diophantine reformulation of the case
$G=\mathbb{Z}/2\mathbb{Z}$
of our result). As a further evidence, we exhibit a few finite groups
$G$
for which the above conclusion holds unconditionally for almost all covers of
$\mathbb{P}_{\mathbb{Q}}^{1}$
of group
$G$
. We also introduce a local–global principle for specializations of Galois covers
$f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$
and show that it often fails if
$f$
has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of
$\mathbb{P}_{\mathbb{Q}}^{1}$
. On the other hand, it allows to generate conditionally ‘many’ curves over
$\mathbb{Q}$
failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.
We describe the irreducible components of the jet schemes with origin in the singular locus of a two-dimensional quasi-ordinary hypersurface singularity. A weighted graph is associated with these components and with their embedding dimensions and their codimensions in the jet schemes of the ambient space. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (equivalent to a divisorial valuation on
$\mathbb{A}^{3}$
), that computes the log-canonical threshold of the singularity embedded in
$\mathbb{A}^{3}$
. This provides us with pairs
$X\subset \mathbb{A}^{3}$
whose log-canonical thresholds are not computed by monomial divisorial valuations. Note that for a pair
$C\subset \mathbb{A}^{2}$
, where
$C$
is a plane curve, the log-canonical threshold is always computed by a monomial divisorial valuation (in suitable coordinates of
$\mathbb{A}^{2}$
).
We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (respectively no smaller than)
$\frac{1}{2}$
is K-stable (respectively K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano variety of dimension
$n$
is at least
$1/(n+1)$
(under mild assumptions) and that the moduli space (if it exists) of birationally superrigid Fano varieties is separated.
For any prime number
$p$
and field
$k$
, we characterize the
$p$
-retract rationality of an algebraic
$k$
-torus in terms of its character lattice. We show that a
$k$
-torus is retract rational if and only if it is
$p$
-retract rational for every prime
$p$
, and that the Noether problem for retract rationality for a group of multiplicative type
$G$
has an affirmative answer for
$G$
if and only if the Noether problem for
$p$
-retract rationality for
$G$
has a positive answer for all
$p$
. For every finite set of primes
$S$
we give examples of tori that are
$p$
-retract rational if and only if
$p\notin S$
.
This paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor E of the canonical blow-up BI ℂn of the Bautin ideal I. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs Arc(BI ℂn, E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.
We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away from a codimension-two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt (divisorially log terminal) models along one-dimensional strata.