arXiv subjects

Hamid Abban

Publications and source records attributed to Hamid Abban.

23 records · Page 2Linked to original sources

arXiv2014

On pliability of del Pezzo fibrations and Cox rings

We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne-Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over $\mathbb{P}^1$, into other Mori fibre spaces, using Cox rings and variation of geometric invariant theory. We show that the pliability of these Mori fibre spaces is at least three and they are not rational.

math.AG↗

arXiv2014

An algebraic study of linkages with helical joints

Methods from algebra and algebraic geometry have been used in various ways to study linkages in kinematics. These methods have failed so far for the study of linkages with helical joints (joints with screw motion), because of the presence of some non-algebraic relations. In this article, we explore a delicate reduction of some analytic equations in kinematics to algebraic questions via a theorem of Ax. As an application, we give a classification of mobile closed 5-linkages with revolute, prismatic, and helical joints.

cs.RO↗

arXiv2014

Circle of Sarkisov links on a Fano $3$-fold

For a general Fano $3$-fold of index $1$ in the weighted projective space $\mathbb{P}(1,1,1,1,2,2,3)$ we construct $2$ new birational models that are Mori fibre spaces, in the framework of the so-called Sarkisov program. We highlight a relation between the corresponding birational maps, as a circle of Sarkisov links, visualising the notion of relations (due to Kaloghiros) in Sarkisov program.

math.AG↗

arXiv2014

Singular del Pezzo fibrations and birational rigidity

A known conjecture of Grinenko in birational geometry asserts that a Mori fibre space with the structure of del Pezzo fibration of low degree is birationally rigid if and only if its anticanonical class is an interior point in the cone of mobile divisors. The conjecture is proved to be true for smooth models (with a generality assumption for degree 3). It is speculated that the conjecture holds for, at least, Gorenstein models in degree 1 and 2. In this article, I present a (Gorenstein) counterexample in degree 2 to this conjecture.

math.AG↗

arXiv2010

On non-rigid del Pezzo fibrations of low degree

We consider $\mathbb{P}(1,1,1,2)$ bundles over $\mathbb{P}^1$ and construct hypersurfaces of these bundles which form a degree 2 del Pezzo fibration over $\mathbb{P}^1$ as a Mori fibre space. We classify all such hypersurfaces whose type $\III$ or $\IV$ Sarkisov links pass to a different Mori fibre space. A similar result for cubic surface fibrations over $\mathbb{P}^2$ is also presented.

math.AG↗