Abstract
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 delivers all Mori dream spaces with torus action. We exhibit the example class of ‘general arrangement varieties’ and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.
Highlights
This article contributes to the study of algebraic varieties with torus action
A torus is an algebraic group T isomorphic to the k-fold direct product Tk of the multiplicative group K∗ of the ground field K, which is assumed to be algebraically closed and of characteristic zero
We provide the necessary background and fix our notation on toric geometry and Cox rings
Summary
This article contributes to the study of algebraic varieties with torus action. Here, a torus is an algebraic group T isomorphic to the k-fold direct product Tk of the multiplicative group K∗ of the ground field K, which is assumed to be algebraically closed and of characteristic zero. Every smooth projective general arrangement variety of true complexity two and Picard number two is isomorphic to precisely one of the following varieties X , specified by their Cox ring R(X ), the matrix [w1, . In Remark 9.4, we obtain a similar finiteness feature as observed in [31] in complexity one: all varieties of Theorem 9.1 arise via two elementary contractions and a series of isomorphisms in codimension one from a finite set of smooth projective general arrangement varieties of complexity two having dimension 5 to 8. We list in Theorem 9.5 the smooth truly almost Fano general arrangement varieties of true complexity two and Picard number two, where truly almost Fano means that the anticanonical divisor is semiample but not ample
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