@misc{AwyongHenkScott1996, author = {Awyong, Poh Wah and Henk, Martin and Scott, Paul R.}, title = {Note on lattice-point-free convex bodies}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2546}, number = {SC-96-44}, year = {1996}, abstract = {We prove inequalities relating the inradius of a convex body with interior containing no point of the integral lattice, with the volume or surface area of the body. These inequalities are tight and generalize previous results.}, language = {en} } @misc{HenkWeismantel1996, author = {Henk, Martin and Weismantel, Robert}, title = {On Hilbert bases of polyhedral cones}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2230}, number = {SC-96-12}, year = {1996}, abstract = {For a polyhedral cone \$C=\$ pos \$\{a^1,\dots,a^m\}\subset R^d\$, \$a^i\in Z^d\$, a subset of integral vectors \$H(C)\subset C \cap Z^d\$ is called a Hilbert basis of \$C\$ iff (i) each element of \$C\cap Z^d\$ can be written as a non-negative integer combination of elements of \$H(C)\$ and (ii) \$H(C)\$ has minimal cardinality with respect to all subsets of \$C \cap Z^d\$ for which (i) holds. We show that various problems related to Hilbert bases are hard in terms of computational complexity. However, if the dimension and the number of elements of the Hilbert basis are fixed, a Hilbert basis can always be computed in polynomial time. Furthermore we introduce a (practical) algorithm for computing the Hilbert basis of a polyhedral cone. The finiteness of this method is deduced from a result about the height of a Hilbert basis which, in particular, improves on former estimates.}, language = {en} } @misc{DaisHenk1997, author = {Dais, Dimitrios I. and Henk, Martin}, title = {On a series of Gorenstein cyclic quotient singularities admitting a unique projective crepant resolution.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3084}, number = {SC-97-39}, year = {1997}, abstract = {Let \$G\$ be a finite subgroup of SL\$\left( r,\% {\mathbb{C}}\right) \$. In dimensions \$r=2\$ and \$r=3\$, McKay correspondence provides a natural bijection between the set of irreducible representations of \$G\$ and a cohomology-ring basis of the overlying space of a projective, crepant desingularization of \${\mathbb{C}}^r/G\$. For \$r=2\$ this desingularization is unique and is known to be determined by the Hilbert scheme of the \$G\$\% -orbits. Similar statements (including a method of distinguishing just {\it{one}} among all possible smooth minimal models of \${\mathbb{C}}^3/G\$), are very probably true for all \$G\$'s \$\subset \$ SL\$\left( 3,{\mathbb{C}}\right) \$ too, and recent Hilbert-scheme-techniques due to Ito, Nakamura and Reid, are expected to lead to a new fascinating uniform theory. For dimensions \$r\geq 4 \$, however, to apply analogous techniques one needs extra modifications. In addition, minimal models of \${\mathbb{C}}^r/G\$ are smooth only under special circumstances. \${\mathbb{C}}^4/\left( \hbox{\rm involution}\right) \$, for instance, cannot have any smooth minimal model. On the other hand, all abelian quotient spaces which are c.i.'s can always be fully resolved by torus-equivariant, crepant, projective morphisms. Hence, from the very beginning, the question whether a given Gorenstein quotient space \${\mathbb{C}}\% ^r/G\$, \$r\geq 4\$, admits special desingularizations of this kind, seems to be absolutely crucial.\noindent In the present paper, after a brief introduction to the existence-problem of such desingularizations (for abelian \$G\$'s) from the point of view of toric geometry, we prove that the Gorenstein cyclic quotient singularities of type \[ \frac 1l\,\left( 1,\ldots ,1,l-\left( r-1\right) \right) \] with \$l\geq r\geq 2\$, have a \textit{unique }torus-equivariant projective, crepant, partial resolution, which is full'' iff either \$l\equiv 0\$ mod \$\% \left( r-1\right) \$ or \$l\equiv 1\$ mod \$\left( r-1\right) \$. As it turns out, if one of these two conditions is fulfilled, then the exceptional locus of the full desingularization consists of \$\lfloor\frac{l}{r-1} \rfloor \$ prime divisors, \$\lfloor\frac{l}{r-1} \rfloor -1\$ of which are isomorphic to the total spaces of \${\mathbb{P}}_{{\mathbb{C}}}^1\$-bundles over \${\mathbb{P}}_{{\mathbb{C}}\% }^{r-2}\$. Moreover, it is shown that intersection numbers are computable explicitly and that the resolution morphism can be viewed as a composite of successive (normalized) blow-ups. Obviously, the monoparametrized singularity-series of the above type contains (as its first member'') the well-known Gorenstein singularity defined by the origin of the affine cone which lies over the \$r\$-tuple Veronese embedding of \${\mathbb{P}}_{\mathbb{C}}^{r-1}\$.}, language = {en} } @misc{DaisHausHenk1998, author = {Dais, Dimitrios I. and Haus, Utz-Uwe and Henk, Martin}, title = {On crepant resolutions of 2-parameter series of Gorenstein cyclic quotient singularities}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3559}, number = {SC-98-12}, year = {1998}, abstract = {\noindent An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces \$\Bbb{C}^{r}/G\$ in dimensions \$r\geq 4\$ would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and sufficient conditions under which 2-parameter series of Gorenstein cyclic quotient singularities have torus-equivariant resolutions of this specific sort in all dimensions.}, language = {en} }