Related papers: Rational homogeneous spaces as geometric realizati…

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on…

We investigate the birational geometry (in the sense of Mori's program) of the moduli space of rank 2 semistable parabolic vector bundles on a rational curve. We compute the effective cone of the moduli space and show that all birational…

We recall a higher dimension analog of the classic plane de Jonqui\`eres transformations, as given by Hassanzadeh and Simis. Such a parameterization defines a birational map from $\mathbb{P}^{n-1}$ to a hypersurface in $\mathbb{P}^{n}$, and…

We complete the classification of regular generically free actions of finite groups on del Pezzo surfaces, up to birational equivalence. As a byproduct, we settle several open problems in equivariant birational geometry, e.g., we classify…

The Hessian map is the rational map that sends a homogeneous polynomial to the determinant of its Hessian matrix. We prove that the Hessian map is birational on its image for ternary forms of degree $d\ge 4$, $d\neq 5$, by considering the…

Let $X$ be a projective variety of dimension $r$ over an algebraically closed field. It is proven that two birational embeddings of $X$ in $\P^n$, with $n\geq r+2$ are equivalent up to Cremona transformations of $\P^n$.

We observe that O'Grady's birational maps between moduli of sheaves on an elliptic K3 surface can be interpreted as intermediate wall-crossing (wall-hitting) transformations at the so-called totally semistable walls, studied by Bayer and…

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative…

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative…

We use the theory of bialgebras to provide the algebraic background for state space realization theorems for input-output maps of control systems. This allows us to consider from a common viewpoint classical results about formal state space…

We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…

Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than…

Starting from $\mathbb{C}^*$-actions on complex projective varieties, we construct and investigate birational maps among the corresponding extremal fixed point components. We study the case in which such birational maps are locally…

We will consider an explicit birational map between a quadric and the projective variety X(J) of traceless rank one elements in a simple reduced Jordan algebra J. X(J) is a homogeneous G-variety for the automorphism group G=Aut(J). We will…

Playing off against each other the real and complex structures, we elucidate the local structure of certain representation spaces in the world of Poisson geometry. Particular cases of these spaces arise as moduli spaces of semistable…

We study determinantal Cremona maps, i.e. birational maps whose base ideal is the maximal minors ideal of a given matrix $\Phi$, via the resolution of the polynomials systems defined by $\Phi$. Using convex geometry, this approach leads in…

We construct universal geometric spaces over the real spectrum compactification $\Xi^{\mathrm{RSp}}$ of the character variety $\Xi$ of a finitely generated group $\Gamma$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary…

We construct families of birational involutions on $\mathbb{P}^3$ or a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of…

Given an algebraic structure on the homology of a chain complex, we define its realization space as a Kan complex whose vertices are the structures up to homotopy realizing this structure at the homology level. Our algebraic structures are…

We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on…