Milena Wrobel

Milena Wrobel
Carl von Ossietzky Universität Oldenburg · Department of Mathematics (IfM)

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14
Publications
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Introduction

Publications

Publications (14)
Article
We introduce the notion of intrinsic Grassmannians that generalizes the well-known weighted Grassmannians. An intrinsic Grassmannian is a normal projective variety whose Cox ring is defined by the Plucker ideal $I_{d,n}$ of the Grassmannian $\textrm{Gr}(d,n)$. We give a complete classification of all smooth Fano intrinsic Grassmannians of type $(2,...
Preprint
We introduce the notion of intrinsic Grassmannians which generalizes the well known weighted Grassmannians. An intrinsic Grassmannian is a normal projective variety whose Cox ring is defined by the Pl\"ucker ideal $I_{d,n}$ of the Grassmannian $\mathrm{Gr}(d,n)$. We give a complete classification of all smooth Fano intrinsic Grassmannians of type $...
Preprint
Full-text available
The anticanonical complex generalizes the Fano polytope from toric geometry and has been used to study Fano varieties with torus action so far. We work out the case of complete intersections in toric varieties defined by non-degenerate systems of Laurent polynomials. As an application, we classify the terminal Fano threefolds that are embedded into...
Preprint
Full-text available
We study three-dimensional Fano varieties with $\mathbb{C}^*$-action. Complementing recent results [13], we give classification results in the canonical case, where the maximal orbit quotient is $\mathbb{P}_2$ having a line arrangement of five lines in special position as its critical values.
Article
We give an explicit description of the divisor class groups of rational trinomial varieties. As an application, we relate the iteration of Cox rings of any rational variety with torus action of complexity one to that of a Du Val surface.
Preprint
We investigate degenerations of cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. We work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and leave the question of classifying these degenerations in the degree 3 case as a...
Preprint
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The anticanonical complex has been introduced as a natural generalization of the toric Fano polytope and so far has been succesfully used for the study of varieties with a torus action of complexity one. In the present article we enlarge the area of application of the anticanonical complex to varieties with a torus action of higher complexity, for...
Preprint
We study full rank homogeneous valuations on (multi)-graded domains and ask when they have finite Khovanskii bases. We show that there is a natural reduction from multigraded to simply graded domains. As special cases, we consider projective coordinate rings of rational curves, and almost toric varieties. Our results relate to several problems pose...
Preprint
Full-text available
We give an explicit description of the divisor class groups of rational trinomial varieties. As an application, we relate the iteration of Cox rings of any rational variety with torus action of complexity one to that of a Du Val surface.
Article
Full-text available
Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension coming with a tor...
Article
Full-text available
We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and de...
Article
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We characterize all varieties with a torus action of complexity one that admit iteration of Cox rings.
Article
Full-text available
We consider rational varieties with a torus action of complexity one and extend the combinatorial approach via the Cox ring developed for the complete case in earlier work to the non-complete, e.g. affine, case. This includes in particular a description of all factorially graded affine algebras of complexity one with only constant homogeneous inver...