Klaus Altmann

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We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our theory extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.
Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal , algebraic varieties. Based on the notion of a " proper polyhedral divisor " introduced in earlier work, we develop the concept of a " divisorial fan " and show that these objects encode the equivariant gluing(More)
There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring K[x 1 ,. .. , x n ]. The definition involves Gröbner bases or the action of the algebraic torus (K *) n. We present algorithms for computing the (affine schemes representing) edges in this graph. We study the induced sub-graphs on multigraded Hilbert schemes and on(More)
We show that Gorenstein singularities that are cones over singular Fano varieties provided by so-called flag quivers are smoothable in codimension three. Moreover, we give a precise characterization about the smoothability in codimension three of the Fano variety itself.