## Study of certain families of algebraic varieties endowed with an algebraic group action

L'anneau de Cox d'une variété algébrique (satisfaisant des conditions naturelles) est un invariant très riche. Il est introduit par Cox en 1995 pour l'étude des variétés toriques, puis généralisé aux variétés normales par Arzhantsev, Berchtold et Hausen. Plus tard, Hu et Keel découvrent que les variétés normales dont l'anneau de Cox est de type fin...

Published in Mathematical Notes

Abstract A criterion for the total coordinate space of a trinomial hypersurface to be a hypersurface is found. An algorithm for calculating the Cox ring in explicit form is proposed, and criteria for the total coordinate space to be rational and factorial are obtained.

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, res...

We consider surfaces X defined by plane divisorial valuations v of the quotient field of the local ring R at a closed point p of the projective plane P-2 over an arbitrary algebraically closed field k and centered at R. We prove that the regularity of the cone of curves of X is equivalent to the fact that v is non-positive on Op(2) (P-2 \ L), where...

Published in Expositiones Mathematicae

Published in Open Mathematics

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\bar X \to X$$ of X as a quotient of a vector space $$\bar X$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient p...

Published in Mathematische Zeitschrift

Let X be a smooth Mori dream space of dimension ≥ 4. We show that, if X satisfies a suitable GIT condition which we call small unstable locus, then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Néron–Severi spaces of X and Y, and under this identification every Mori chamber of Y is a unio...

Published in Advances in Mathematics

Published in Mathematical Notes

The generalized Cox construction associates with an algebraic variety a remarkable invariant—its total coordinate ring, or Cox ring. In this note, we give a new proof of the factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on the notion of graded factoriality. We show that i...