Related papers: The birational geometry of GIT quotients
In this note we discuss some arithmetic and geometric questions concerning self maps of projective algebraic varieties.
The Witt group of a smooth curve over a real closed field is explicitely calculated. The method uses a comparison theorem between the graded Witt group and the etale cohomology groups. In the second part of the paper, the torsion Picard…
The moduli space $\cM_g$ of nonsingular projective curves of genus $g$ is compactified into the moduli $\bcM_g$ of Deligne-Mumford stable curves of genus $g$. We compactify in a similar way the moduli space of abelian varieties by adding…
The Geometric Algebra Transformer (GATr) is a versatile architecture for geometric deep learning based on projective geometric algebra. We generalize this architecture into a blueprint that allows one to construct a scalable transformer…
Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a…
We study GIT stability of divisors in products of projective spaces. We first construct a finite set of one-parameter subgroups sufficient to determine the stability of the GIT quotient. In addition, we characterise all maximal orbits of…
In a previous paper, the author and David Swinarski constructed the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). That paper required the base field to be the complex numbers, a restriction which…
In this paper we construct new derived invariants with integral coefficients using the theory of motifs, and give several applications. Specifically, we obtain the following results: For complex algebraic surfaces, we prove that certain…
For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the…
In this thesis we study toric degenerations of projective varieties. We compare different constructions to understand how and why they are related as s first step towards developing a global framework. In focus are toric degenerations…
Let $K$ be an algebraically closed field of arbitrary characteristic and let $X$ be an irreducible projective variety over $K$. Let $G\subseteq\text{Bir}(X)$ be a bounded-degree subgroup. We prove that there exists an irreducible projective…
A gauge-invariant formulation for the gravitational wave equations is presented. Using this approach, weak, plane wave solutions in a vacuum are derived in various theories. These include general relativity with two modes of polarization…
Let $X$ be a smooth complex projective algebraic variety. Let $\mathcal{G}$ be a $G$-banded gerbe with $G$ a finite abelian group. We prove an exact formula expressing genus $g$ orbifold Gromov-Witten invariants of $\mathcal{G}$ in terms of…
Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented…
Geometric problems are usually formulated by means of (exterior) differential systems. In this theory, one enriches the system by adding algebraic and differential constraints, and then looks for regular solutions. Here we adopt a dual…
We present an overview of the role of generating functions in quantum mechanical contexts, mainly in the modern theory of polarization and in the study of quantum phase transitions. Generating functions enable the derivation of moments and…
For G a complex reductive group and X a smooth projective or convex quasi-projective polarized G-variety we construct a formal map in quantum K-theory from the equivariant quantum K-theory $QK^G(X)$ to the quantum K-theory of the git…
Although there is no natural internal product for hermitian forms over an algebra with involution of the first kind, we describe how to multiply two $\varepsilon$-hermitian forms to obtain a quadratic form over the base field. This allows…
It is well known that in a generally covariant gravitational theory the choice of spacetime scalars as coordinates yields phase-space observables (or "invariants"). However their relation to the symmetry group of diffeomorphism…
The theory of algebras with polynomial identities has developed significantly, with special attention devoted to the classification of varieties according to the asymptotic behavior of their codimension sequences. This sequence is a…