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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.
In a previous paper, we discussed Frobenius-projective structures on projective smooth curves in positive characteristic and established a relationship between pseudo-coordinates and Frobenius-indigenous structures by means of Frobenius-projective structures. In the present paper, we discuss an “affine version” of this study of Frobenius-projective structures. More specifically, we discuss Frobenius-affine structures and establish a similar relationship between Tango functions and Frobenius-affine-indigenous structures by means of Frobenius-affine structures. Moreover, we also consider a relationship between these objects and Tango curves.
We investigate a novel geometric Iwasawa theory for
${\mathbf Z}_p$
-extensions of function fields over a perfect field k of characteristic
$p>0$
by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if
$\cdots \to X_2 \to X_1 \to X_0$
is the tower of curves over k associated with a
${\mathbf Z}_p$
-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of
$X_n$
as
$n\rightarrow \infty $
. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of
$X_n$
equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the
$k[V]$
-module structure of the space
$M_n:=H^0(X_n, \Omega ^1_{X_n/k})$
of global regular differential forms as
$n\rightarrow \infty .$
For example, for each tower in a basic class of
${\mathbf Z}_p$
-towers, we conjecture that the dimension of the kernel of
$V^r$
on
$M_n$
is given by
$a_r p^{2n} + \lambda _r n + c_r(n)$
for all n sufficiently large, where
$a_r, \lambda _r$
are rational constants and
$c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$
is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on
${\mathbf Z}_p$
-towers of curves, and we prove our conjectures in the case
$p=2$
and
$r=1$
.
We consider an analogue of Kontsevich’s matrix Airy function where the cubic potential
$\textrm{Tr}(\Phi^3)$
is replaced by a quartic term
$\textrm{Tr}\!\left(\Phi^4\right)$
. Cumulants of the resulting measure are known to decompose into cycle types for which a recursive system of equations can be established. We develop a new, purely algebraic geometrical solution strategy for the two initial equations of the recursion, based on properties of Cauchy matrices. These structures led in subsequent work to the discovery that the quartic analogue of the Kontsevich model obeys blobbed topological recursion.
In this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative Korteweg-de Vries hierarchy, which we introduced in a previous work. The specialisation of this conjecture to the top degree part of Pixton’s class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the double ramification/Dubrovin–Zhang equivalence conjecture. We also provide several independent computational checks in support of it.
We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne–Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.
We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.
Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree d and dimension r on a general curve of genus g satisfying prescribed incidence conditions at n points. We determine these numbers completely for linear series of arbitrary dimension when d is sufficiently large, and for all d when either
$r=1$
or
$n=r+2$
. Our formulas generalise and give new proofs of recent results of Tevelev and of Cela, Pandharipande and Schmitt.
Let p be a prime number. In the present paper, we prove that the moduli of hyperbolic curves of genus $0$ over an algebraic closure of the field of p-adic numbers may be completely determined by their tempered fundamental groups.
We study the p-rank stratification of the moduli space of cyclic degree
$\ell $
covers of the projective line in characteristic p for distinct primes p and
$\ell $
. The main result is about the intersection of the p-rank
$0$
stratum with the boundary of the moduli space of curves. When
$\ell =3$
and
$p \equiv 2 \bmod 3$
is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type
$(r,s)$
, and p-rank f satisfying the clear necessary conditions.
For X a smooth projective variety and
$D=D_1+\dotsb +D_n$
a simple normal crossing divisor, we establish a precise cycle-level correspondence between the genus
$0$
local Gromov–Witten theory of the bundle
$\oplus _{i=1}^n \mathcal {O}_X(-D_i)$
and the maximal contact Gromov–Witten theory of the multiroot stack
$X_{D,\vec r}$
. The proof is an implementation of the rank-reduction strategy. We use this point of view to clarify the relationship between logarithmic and orbifold invariants.
We prove that
$164\, 634\, 913$
is the smallest positive integer that is a sum of two rational sixth powers, but not a sum of two integer sixth powers. If
$C_{k}$
is the curve
$x^{6} + y^{6} = k$
, we use the existence of morphisms from
$C_{k}$
to elliptic curves, together with the Mordell–Weil sieve, to rule out the existence of rational points on
$C_{k}$
for various k.
We prove a formula, which, given a principally polarized abelian variety $(A,\lambda )$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron–Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,\lambda )$.
We discuss the
$\ell $
-adic case of Mazur’s ‘Program B’ over
$\mathbb {Q}$
: the problem of classifying the possible images of
$\ell $
-adic Galois representations attached to elliptic curves E over
$\mathbb {Q}$
, equivalently, classifying the rational points on the corresponding modular curves. The primes
$\ell =2$
and
$\ell \ge 13$
are addressed by prior work, so we focus on the remaining primes
$\ell = 3, 5, 7, 11$
. For each of these
$\ell $
, we compute the directed graph of arithmetically maximal
$\ell $
-power level modular curves
$X_H$
, compute explicit equations for all but three of them and classify the rational points on all of them except
$X_{\mathrm {ns}}^{+}(N)$
, for
$N = 27, 25, 49, 121$
and two-level
$49$
curves of genus
$9$
whose Jacobians have analytic rank
$9$
.
Aside from the
$\ell $
-adic images that are known to arise for infinitely many
${\overline {\mathbb {Q}}}$
-isomorphism classes of elliptic curves
$E/\mathbb {Q}$
, we find only 22 exceptional images that arise for any prime
$\ell $
and any
$E/\mathbb {Q}$
without complex multiplication; these exceptional images are realised by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on
$X_{\mathrm {ns}}^+(\ell )$
with
$\ell \ge 19$
, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the
$\ell $
-adic images of Galois for any elliptic curve over
$\mathbb {Q}$
.
In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on
$\Gamma _1(N)$
are of
$\operatorname {GL}_2$
-type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of
$X_H$
.
We propose an intersection-theoretic method to reduce questions in genus 0 logarithmic Gromov–Witten theory to questions in the Gromov–Witten theory of smooth pairs, in the presence of positivity. The method is applied to the enumerative geometry of rational curves with maximal contact orders along a simple normal crossings divisor and to recent questions about its relationship to local curve counting. Three results are established. We produce counterexamples to the local-logarithmic conjectures of van Garrel–Graber–Ruddat and Tseng–You. We prove that a weak form of the conjecture holds for product geometries. Finally, we explicitly determine the difference between local and logarithmic theories, in terms of relative invariants for which efficient algorithms are known. The polyhedral geometry of the tropical moduli of maps plays an essential and intricate role in the analysis.
Motivated by the problem of finding algebraic constructions of finite coverings in commutative algebra, the Steinitz realization problem in number theory and the study of Hurwitz spaces in algebraic geometry, we investigate the vector bundles underlying the structure sheaf of a finite flat branched covering. We prove that, up to a twist, every vector bundle on a smooth projective curve arises from the direct image of the structure sheaf of a smooth, connected branched cover.
We study the Chow ring of the moduli stack
$\mathfrak {M}_{g,n}$
of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space
$\overline {\mathcal {M}}_{g,n}$
of stable curves. This paper provides foundations for the paper [BS21].
In the appendix (jointly with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of
$\mathfrak {M}_{g,n}$
and is important in its own right. We also develop operational Chow groups for algebraic stacks.
We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel–Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition and prove that its global degree is
$2^{g-1}$
. Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel–Prym map is
$2^{g-1}$
as well.
We interpret the degrees which arise in Tevelev’s study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalisation). We find exact solutions which specialise to Tevelev’s formula in his cases and connect to the projective geometry of lines and Castelnuovo’s classical count of
$g^1_d$
’s in other cases. For almost all values, the calculation of the two parameter generalisation of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.
Using the theory of ${\mathbf {FS}} {^\mathrm {op}}$ modules, we study the asymptotic behavior of the homology of ${\overline {\mathcal {M}}_{g,n}}$, the Deligne–Mumford compactification of the moduli space of curves, for $n\gg 0$. An ${\mathbf {FS}} {^\mathrm {op}}$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of ${\overline {\mathcal {M}}_{g,n}}$ the structure of an ${\mathbf {FS}} {^\mathrm {op}}$ module and bound its degree of generation. As a consequence, we prove that the generating function $\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$ is rational, and its denominator has roots in the set $\{1, 1/2, \ldots, 1/p(g,i)\},$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$. We also obtain restrictions on the decomposition of the homology of ${\overline {\mathcal {M}}_{g,n}}$ into irreducible $\mathbf {S}_n$ representations.