Searcharxiv⌕ Search

arXiv subjects

Catriona Maclean

Publications and source records attributed to Catriona Maclean.

12 recordsLinked to original sources
arXiv2024

Cohomology classes of complex approximable algebras

Huayi Chen introduces the notion of an approximable graded algebra, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asked if any such algebra is the graded ring of a big line bundle on a projective variety. This was proved to be false in a previous paper of the author's, who subsequently proved that any such algebra is associated to an infinite Weil divisor. In this paper, we show that over the complex numbers, this infinite Weil divisor necessarily has finite cohomology class.

math.AG↗
arXiv2019

Concave transforms of filtrations and rationality of Seshadri constants

We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton--Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded algebra is the algebra of sections of a concrete line bundle in higher dimension. We use this description to give a rationality criterion for certain Seshadri constants. Along the way we introduce Newton--Okounkov bodies of abstract graded semigroups and determine conditions for their slices to be Newton--Okounkov bodies of subsemigroups.

math.AG↗
arXiv2017

Classifying approximable algebras

Approximable algebras were defined by Chen in his proof of the Fujita theorem in the arithmetic context. These were shown to not be necessarily subalgebras of section rings of big line bundles in a previous prepreint of the author. Here, we show that whilst approximable algebras are not necessarily subalgebras of section rings of big line bundles, they are necessarily subalgebras of section rings of infinite sums of Weil divisors.

math.AG↗
arXiv2017

Approximable algebras and a question of Chen

In his article "Arithmetic Fujita Approximations", Huayi Chen introduces the notion of an approximable graded algebra and asks if any such algebra is a subalgebra of the graded section ring of a big line bundle on an algebraic variety. We give a counter-example showing this is not the case.

math.AG↗
arXiv2015

Vanishing sequences and Okounkov bodies

We define and study the vanishing sequence along a real valuation of sections of a line bundle on a projective variety. Building on previous work of the first author with Huayi Chen, we prove an equidistribution result for vanishing sequences of large powers of a big line bundle, and study the limit measure. In particular, the latter is described in terms of restricted volumes for divisorial valuations. We also show on an example that the associated concave function on the Okounkov body can be discontinuous at boundary points.

math.AG↗
arXiv2013

Distribution of logarithmic spectra of the equilibrium energy

Let $L$ be a big invertible sheaf on a complex projective variety, equipped with two continuous metrics. We prove that the distribution of the eigenvalues of the transition matrix between the $L^2$ norms on $H^0(X,nL)$ with respect to the two metriques converges (in law) as $n$ goes to infinity to a Borel probability measure on $\mathbb R$. This result can be thought of as a generalization of the existence of the energy at the equilibrium as a limit, or an extension of Berndtsson's results to the more general context of graded linear series and a more general class of line bundles. Our approach also enables us to obtain a $p$-adic analogue of our main result.

math.MG↗
arXiv2012

Volume functions of linear series

The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole Néron--Severi space, thus giving rise to a basic invariant of the underlying projective variety. Analogously, one can also define the volume function of a possibly non-complete multigraded linear series. In this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties. In the multigraded setting, relying on the work of Lazarsfeld and Mustaţă (2009) on Okounkov bodies, we show that any continuous, homogeneous, and log-concave function appears as the volume function of a multigraded linear series. By contrast we show that there exists countably many functions which arise as the volume functions of projective varieties. We end the paper with an example, where the volume function of a projective variety is given by a transcendental formula, emphasizing the complicated nature of the volume in the classical case.

math.AG↗
arXiv2010

Convex bodies appearing as Okounkov bodies of divisors

Based on the work of Okounkov (\cite{Ok96}, \cite{Ok03}), Lazarsfeld and Musta\c t\u a (\cite{LM08}) and Kaveh and Khovanskii (\cite{KK08}) have independently associated a convex body, called the Okounkov body, to a big divisor on a smooth projective variety with respect to a complete flag. In this paper we consider the following question: what can be said about the set of convex bodies that appear as Okounkov bodies? We show first that the set of convex bodies appearing as Okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable. We then give a complete characterisation of the set of convex bodies that arise as Okounkov bodies of $\R$-divisors on smooth projective surfaces. Such Okounkov bodies are always polygons, satisfying certain combinatorial criteria. Finally, we construct two examples of non-polyhedral Okounkov bodies. In the first one, the variety we deal with is Fano and the line bundle is ample. In the second one, we find a Mori dream space variety such that under small perturbations of the flag the Okounkov body remains non-polyhedral.

math.AG↗
arXiv2008

Zariski decomposition of b-divisors

Based on a recent work by Thomas Bauer reproving the existence of Zariski decompositions for surfaces, we construct a b-divisorial analogue of Zariski decomposition in all dimensions.

math.AG↗
arXiv2005

Explicit Noether-Lefschetz for arbitrary threefolds

We study the Noether-Lefschetz locus of a very ample line bundle L on an arbitrary smooth threefold Y. Building on results of Green, Voisin and Otwinowska, we give explicit bounds, depending only on the Castelnuovo-Mumford regularity properties of L, on the codimension of the components of the Noether-Lefschetz locus of |L|.

math.AG↗