We give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy (arXiv:1003.5217). We actually prove not only the original algorithm but also a speed-up version of it. Our proof is independent from (in fact appeared earlier on the arXiv than) the proof by H. Roschy and T. Rahn (arXiv:1006.2392), and has several advantages such as being shorter and cleaner and can also settle the additional conjecture on "Serre duality for Betti numbers" which was raised but unresolved in arXiv:1006.2392.Comment: 9 pages. Theorem 1.1 and Corollary 1.2 improved; Abstract and Introduction modified; References updated. To appear in Journal of Mathematical Physic

Given a big divisor D on a normal complex projective variety X, we show that the restricted volume of D along a very general complete-intersection curve C ⊂ X can be read off from the Okounkov body of D with respect to an admissible flag containing C. From this we deduce that if two big divisors D 1 and D 2 on X have the same Okounkov body with respect to every admissible flag, then D 1 and D 2 are numerically equivalent.

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 union of some Mori chambers of X , and the nef cone of Y is the same as the nef cone of X . This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking "Mori dream hypersurfaces" of an ambient Mori dream space, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly.Keywords Mori dream space · Lefschetz theorem · Nef cone · Mori chamber · m-neighborly fan · Cox ring · GIT quotient IntroductionThe main purpose of this paper is to prove an analog of the Lefschetz hyperplane theorem for Mori dream spaces.Let X be a smooth complex projective variety, and let N 1 (X ) be the group of numerical equivalence classes of line bundles on X . Recall from [17] that X is called a Mori dream space if Pic(X ) Q = N 1 (X ) Q (equivalently H 1 (X, O X ) = 0), and X has a finitely generated Cox ring (Definition 1.2). As the name might suggest, Mori dream spaces are very special varieties on which Mori theory works extremely well (see the nice survey article of Hu [16]). On the other hand, not many classes of examples of them are known. It has been understood for a while that toric varieties are Mori dream spaces; indeed their Cox rings are polynomial

Abstract. Fix nonzero ideal sheaves a 1 , . . . , ar and b on a normal Q-Gorenstein complex variety X. For any positive real numbers α and β, we construct a resolution of the multiplier ideal J ((a 1 + · · · + ar) α b β ) by sheaves that are direct sums of multiplier ideals J (aThe resolution is cellular, in the sense that its boundary maps are encoded by the algebraic chain complex of a regular CW-complex. The CW-complex is naturally expressed as a triangulation ∆ of the simplex of nonnegative real vectors λ ∈ R r with P r i=1 λ i = α. The acyclicity of our resolution reduces to that of a cellular free resolution, supported on ∆, of a related monomial ideal. Our resolution implies the multiplier ideal sum formula

Let U n (F q 2 ) be the n-dimensional unitary group over the finite field F q 2 . In this paper, we find explicit generators and relations for the ring of invariants of U n (F q 2 ), and prove that it is a complete intersection. IntroductionLet V be an n-dimensional vector space over the finite field F q 2 , and H a non-degenerate hermitian form on V (recall that for a hermitian form to exist the order of the base field must be a square). It is well known that up to isomorphisms, there is essentially only one non-degenerate hermitian form over a finite field, and with a suitable choice of basis for V , we may assume that H = x q+1 1 + · · · + x q+1 n . The unitary group U n (F q 2 ) is by definition the subgroup of GL(V ) consisting of those linear transformations which preserve the hermitian form H . In this paper, we find explicit generators and relations for the ring of invariants F q 2 [V ] U n (F q 2 ) , and prove that it is a complete intersection. With some adaptations, our method can also apply to the finite orthogonal groups and prove that their invariant rings are also complete intersections [8].There has long been an interest in finding the invariants of classical groups over finite fields.