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Chapter III : Introduction aux théorèmes de Hilbert-Samuel arithmétiques
Le but de ce chapitre est d’expliquer quelques théorèmes de type Hilbert-Samuel, qui étudient le comportement asymptotique d’un système linéaire...
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Valuative Invariants with Higher Moments
In this article we introduce a family of valuative invariants defined in terms of the p -th moment of the expected vanishing order. These invariants...
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From Convex Geometry of Certain Valuations to Positivity Aspects in Algebraic Geometry
A few years ago Okounkov associated a convex set (Newton–Okounkov body) to a divisor, encoding the asymptotic vanishing behaviour of all sections of...
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Charging solid partitions
Solid partitions are the 4D generalization of the plane partitions in 3D and Young diagrams in 2D, and they can be visualized as stacking of 4D...
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C-Robin functions and applications
We continue the study in [1] in the setting of pluripotential theory arising from polynomials associated to a convex body C in (ℝ + ) d . Here we discuss C ...
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Adelic line bundles on arithmetic varieties
In this chapter, we fix a proper adelic curve \(S=(K,(\varOmega ,\mathcal A,\nu ),\phi )\) .
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A review of geometrically defined functions on Newton–Okounkov bodies
This note is a presentation of two functions carrying geometric information recently defined on the Newton–Okounkov body by Boucksom and Chen (Compos...
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Transfinite Diameter with Generalized Polynomial Degree
We prove a Chebyshev transform formula for a notion of (weighted) transfinite diameter that is defined using a generalized notion of polynomial...
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A Guided Tour to Normalized Volume
This is a survey on the recent theory on minimizing the normalized volume function attached to any klt singularities.
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Newton–Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases
A Newton–Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is...
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On subfiniteness of graded linear series
Hilbert’s fourteenth problem studies the finite generation property of the intersection of an integral algebra of finite type with a subfield of the...
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Algebras Defined by Minors
In Chap. 4 we have studied the Gröbner deformations of determinantal ideals defined by their initial...
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The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope
Regular semisimple Hessenberg varieties are subvarieties of the flag variety Flag(ℂ n ) arising naturally at the intersection of geometry,...
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NEWTON–OKOUNKOV POLYTOPES OF FLAG VARIETIES
We compute the Newton–Okounkov bodies of line bundles on the complete ag variety of GL n for a geometric valuation coming from a ag of translated...
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FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS
We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an...
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Algebraic volumes of divisors
The volume of a Cartier divisor on a projective variety is a nonnegative real number that measures the asymptotic growth of sections of multiples of...
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Cones from quantum groups to tropical flag varieties
We relate quantum degree cones, parametrizing PBW degenerations of quantized envelo** algebras, to (negative tight monomial) cones introduced by...
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Local Okounkov bodies and limits in prime characteristic
This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we...
