https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

Elements of mathematics

/pdf/classical-algebraic-geometry-3wzv0dreho.pdf

Classical Algebraic Geometry

Introduction.- Birational Geometry of Surfaces.- Logarithmic Category.- Overview of Mori's Program.- Singularities.- Vanishing Theorems.- Base Point Freeness of Linear Systems.- Cone Theorem.- Contraction Theorem.- Flip.- Cone Theorem Revisited.- Logarithmic Mori's Program.- Birational Relations among Minimal Models.- Birational Relations among Mori Fiber Spaces.- Birational Geometry of Toric Varieties.

Introduction to the Mori Program

We review previous work of (mainly) Koelman, Haase and Schicho, and Poonen and Rodriguez-Villegas on the dual operations of (i) taking the interior hull and (ii) moving out the edges of a two-dimensional lattice polygon. We show how the latter operation naturally gives rise to an algorithm for enumerating lattice polygons by their genus. We then report on an implementation of this algorithm, by means of which we produce the list of all lattice polygons (up to equivalence) whose genus is contained in {1,…,30}. In particular, we obtain the number of inequivalent lattice polygons for each of these genera. As a byproduct, we prove that the minimal possible genus for a lattice 15-gon is 45.

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Moving Out the Edges of a Lattice Polygon

First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.

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Lattice polygons and families of curves on rational surfaces

Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Su...

https://epubs.siam.org/doi/pdf/10.1137/17M1118312

The Number of Realizations of a Laman Graph

Minimal families of curves on surfaces

In this paper, we consider the classification of singularities [P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction. I, II, III, Proc. Camb. Philos. Soc.30 (1934) 453–491] and real structures [C. T. C. Wall, Real forms of smooth del Pezzo surfaces, J. Reine Angew. Math.1987(375/376) (1987) 47–66, ISSN 0075-4102] of weak Del Pezzo surfaces from an algorithmic point of view. It is well-known that the singularities of weak Del Pezzo surfaces correspond to root subsystems. We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces are also represented by root subsystems. We present an algorithm which computes the classification of real structures. This leads to an alternative proof of the known classification for Del Pezzo surfaces and extends this classification to singular weak Del Pezzo surfaces. As an application we classify families of real conics on cyclides.

Algorithms for singularities and real structures of weak del pezzo surfaces

We present algorithms for reconstructing, up to unavoidable projective automorphisms, surfaces with ordinary singularities in three-dimensional space starting from their silhouette, or “apparent co...

/pdf/reconstruction-of-surfaces-with-ordinary-singularities-from-2of2zhyona.pdf

Niels Lubbes

Papers

Lattice polygons and families of curves on rational surfaces

The Number of Realizations of a Laman Graph

Minimal families of curves on surfaces

Algorithms for singularities and real structures of weak del pezzo surfaces

Reconstruction of Surfaces with Ordinary Singularities from Their Silhouettes