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From Commutative Algebra to Algebraic Geometry

Abstract

We focus on the description of the automorphism group Γ of a Clifford-like parallelism ∥ on a 3-dimensional projective double space (ℙ(HF), ∥, ∥r) over a quaternion skew field H (of any characteristic). We compare Γ with the automorphism group Γ of the left parallelism ∥, which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose ∥ in such a way that Γ is either properly contained in Γ or coincides with Γ even though ∥ ≠ ∥.

Abstract

We study birational projective models of 𝓜2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜 2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M 2,2(𝓩).

Abstract

For a smooth manifold X equipped with a volume form, let 𝓛0 (X) be the Lie algebra of volume preserving smooth vector fields on X. Lichnerowicz proved that the abelianization of 𝓛0 (X) is a finite-dimensional vector space, and that its dimension depends only on the topology of X. In this paper we provide analogous results for some classical examples of non-singular complex affine algebraic varieties with trivial canonical bundle, which include certain algebraic surfaces and linear algebraic groups. The proofs are based on a remarkable result of Grothendieck on the cohomology of affine varieties, and some techniques that were introduced with the purpose of extending the Andersén–Lempert theory, which was originally developed for the complex spaces ℂn, to the larger class of Stein manifolds that satisfy the density property.

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Abstract

Let S be a surface of genus g at least 2. A representation ρ:π1SPSL2R is said to be purely hyperbolic if its image consists only of hyperbolic elements along with the identity. We may wonder under which conditions such representations arise as the holonomy of a branched hyperbolic structure on S. In this work we characterise them completely, giving necessary and sufficient conditions.

Abstract

Let Q be a simply connected manifold. We show that every exact Lagrangian cobordism between compact, exact Lagrangians in T * Q is an h-cobordism. This is a corollary of the Abouzaid–Kragh Theorem.

Abstract

We show that the Grothendieck group associated to integral polytopes in ℝn is free-abelian, by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sending a polytope to its induced seminorm on the dual of ℝn.

Abstract

K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to 30 vertices. Their number is 36 297 333. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.