To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure firstname.lastname@example.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Let X be a non-singular irreducible complex projective curve of genus g ≥ 2. The concept of stability of coherent systems over X depends on a positive real parameter α, given then a (finite) family of moduli spaces of coherent systems. We use (t, ℓ)-stability to prove the existence of coherent systems over X that are α-stable for all allowed α > 0.
Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel–Jacobi map. This breaks down over the boundary since the Abel–Jacobi map fails to extend. We construct a ‘universal’ resolution of the Abel–Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.
We prove constancy of Newton polygons of all convergent
-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of proper smooth families of curves over Abelian varieties. More generally, we prove the isotriviality over projective smooth varieties on which any convergent
-isocrystal has constant Newton polygons.
We study the dynamics of a singular holomorphic vector field at
. Using the associated flow and its pullback to the blow-up manifold, we provide invariants relating the vector field, a non-invariant analytic branch of curve, and the deformation of this branch by the flow. This leads us to study the conjugacy classes of singular branches under the action of holomorphic flows. In particular, we show that there exists an analytic class that is not complete, meaning that there are two elements of the class that are not analytically conjugated by a local biholomorphism embedded in a one-parameter flow. Our techniques are new and offer an approach dual to the one used classically to study singularities of holomorphic vector fields.
be a smooth projective Fano variety over the complex numbers. We study the moduli space of rational curves on
using the perspective of Manin’s conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on
. We propose a geometric Manin’s conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.
We prove the existence of a smooth and non-degenerate curve
, and general moduli for all
$d\geqslant (n-3)\lceil g/2\rceil +n+3$
. It was proved by C. Walter that, for
, the inequality
is a necessary condition for the existence of a curve with
We generalize the Cohen–Lenstra heuristics over function fields to étale group schemes
(with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg–Venkatesh–Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of
, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian
-groups in cases where
$\ell \mid q-1$
; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg–Venkatesh–Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.
F. Cukierman asked whether or not for every smooth real plane curve
of even degree
there exists a real line
has no real points. We show that the answer is yes if
or 4 and no if
We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.
We prove that if
is a reflexive smooth plane curve of degree
defined over a finite field
, then there is an
transversely. We also prove the same result for non-reflexive curves of degree
be a smooth projective curve of genus
over an algebraically closed field
. We show that for any integers
, there exists a maximally Frobenius destabilised stable vector bundle of rank
if and only if
be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by
on Teichmüller space descends to a multivalued self-map—a Hurwitz correspondence
—of the moduli space
. We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees. We show that the sequence of dynamical degrees of
is always non-increasing and that the behavior of this sequence is constrained by the behavior of
at and near points of its post-critical set.
We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.
This paper contains two results on Hodge loci in
. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibers and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in
. It is proved that the image under the period map of a divisor in
is not contained in a proper totally geodesic subvariety of
. It follows that a Hodge locus in
has codimension at least 2.
The motivic Hilbert zeta function of a variety
is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points on
. In this paper, the motivic Hilbert zeta function of a reduced curve is shown to be rational.
We investigate the connection between Osserman limit series (on curves of pseudocompact type) and Amini–Baker limit linear series (on metrized complexes with corresponding underlying curve) via a notion of pre-limit linear series on curves of the same type. Then, applying the smoothing theorems of Osserman limit linear series, we deduce that, fixing certain metrized complexes, or for certain types of Amini–Baker limit linear series, the smoothability is equivalent to a certain “weak glueing condition”. Also for arbitrary metrized complexes of pseudocompact type the weak glueing condition (when it applies) is necessary for smoothability. As an application we confirm the lifting property of specific divisors on the metric graph associated with a certain regular smoothing family, and give a new proof of a result of Cartright, Jensen, and Payne for vertex-avoiding divisors, and generalize it for divisors of rank one in the sense that, for the metric graph, there could be at most three edges (instead of two) between any pair of adjacent vertices.
We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine
-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.
In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight
be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck’s étale fundamental group
. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori’s profinite fundamental group scheme
, and give a partial answer to it.
We study Higgs bundles over an elliptic curve with complex reductive structure group, describing the (normalisation of) its moduli spaces and the associated Hitchin fibration. The case of trivial degree is covered by the work of Thaddeus in 2001. Our arguments are different from those of Thaddeus and cover arbitrary degree.