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The complete classification of hexapods - also known as Stewart Gough platforms - of mobility one is still open. To tackle this problem, we can associate to each hexapod of mobility one an algebraic curve, called the configuration curve. In this paper we establish an upper bound for the degree of this curve, assuming the hexapod is general enough. Moreover, we provide a construction of hexapods with curves of maximal degree, which is based on liaison, a technique used in the theory of algebraic curves.Comment: 40 pages, 6 figure

Fano manifolds of index n-1 and the cone conjecture

The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair $(X, \Delta)$ have finite, rational polyhedral fundamental domains. Let $Z$ be an $n$-dimensional Fano manifold of index $n-1$ such that $-K_Z = (n-1) H$ for an ample divisor $H$. Let $\Gamma$ be the base locus of a general $(n-1)$-dimensional linear system $V \subset |H|$. In this paper, we verify the Morrison-Kawamata cone conjecture for the blow-up of $Z$ along $\Gamma$.Comment: 30 page

Algebraic Geometry: Birational Classification, Derived Categories, and Moduli Spaces

The workshop covered a number of active areas of research in algebraic geometry with a focus on derived categories, moduli spaces (of varieties and sheaves) and birational geometry (often in positive characteristic) and their interactions. Special emphasis was put on hyperkähler manifolds and singularity theory

On the Sylvester-Gallai and the orchard problem for pseudoline arrangements

We study a non-trivial extreme case of the orchard problem for $12$ pseudolines and we provide a complete classification of pseudoline arrangements having $19$ triple points and $9$ double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of B\"or\"oczky. Since Melchior's inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Gr\"unbaum's problems. We formulate some open problems which involve our new examples of line arrangements.Comment: 5 figures, 11 pages, to appear in Periodica Mathematica Hungaric

Enhanced Gauge Symmetry in Type II String Theory

We show how enhanced gauge symmetry in type II string theory compactified on a Calabi--Yau threefold arises from singularities in the geometry of the target space. When the target space of the type IIA string acquires a genus $g$ curve $C$ of $A_{N-1}$ singularities, we find that an $SU(N)$ gauge theory with $g$ adjoint hypermultiplets appears at the singularity. The new massless states correspond to solitons wrapped about the collapsing cycles, and their dynamics is described by a twisted supersymmetric gauge theory on $C\times \R^4$. We reproduce this result from an analysis of the $S$-dual $D$-manifold. We check that the predictions made by this model about the nature of the Higgs branch, the monodromy of period integrals, and the asymptotics of the one-loop topological amplitude are in agreement with geometrical computations. In one of our examples we find that the singularity occurs at strong coupling in the heterotic dual proposed by Kachru and Vafa.Comment: 43 pages using harvmac's `big' option. (Minor correction concerning magnetically charged states.

Duality and polyhedrality of cones for Mori dream spaces

Our goal is twofold. On one hand we show that the cones of divisors ample in codimension $k$ on a Mori dream space are rational polyhedral. On the other hand we study the duality between such cones and the cones of $k$-moving curves by means of the Mori chamber decomposition of the former. We give a new proof of the weak duality property (already proved by Payne and Choi) and we exhibit an interesting family of examples for which strong duality holds.Comment: 20 pages, 1 figur

Fano manifolds of index n-1 and the cone conjecture

The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair (X,Δ) have finite, rational polyhedral fundamental domains. Let Z be an n-dimensional Fano manifold of index n-1 such that -KZ=(n-1)H for an ample divisor H. Let Γ be the base locus of a general (n-1)-dimensional linear system V ⊂/H/. In this paper, we verify the Morrison-Kawamata cone conjecture for the blowup of Z along Γ. © 2013 The Author(s). Published by Oxford University Press. All rights reserved