arXiv subjects

Robin Guilbot

Publications and source records attributed to Robin Guilbot.

4 recordsLinked to original sources

arXiv2026

Calabi-Yau complete intersections associated to good pairs of generalized nef partitions

We introduce the notion of good pair of generalized nef partitions to describe Calabi-Yau complete intersections in Q-Fano toric varieties whose equations do not necessarily have maximal Newton polytopes. Moreover, we define a natural duality between them which generalizes Batyrev-Borisov mirror duality and allows to define a generalization of Berglund-H\"ubsch-Krawitz duality to quasismooth complete intersections.

math.AG↗

arXiv2018

Quasismoooth hypersurfaces in toric varieties

We provide a combinatorial characterization of monomial linear systems on toric varieties whose general member is quasismooth. This is given both in terms of the Newton polytope and in terms of the matrix of exponents of a monomial basis.

math.AG↗

arXiv2015

Families of Calabi-Yau hypersurfaces in $\mathbb Q$-Fano toric varieties

We provide a sufficient condition for a general hypersurface in a $\mathbb Q$-Fano toric variety to be a Calabi-Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund-H\"ubsch-Krawitz construction in case the ambient is a $\mathbb Q$-Fano toric variety with torsion free class group and the defining polynomial is not necessarily of Delsarte type. Finally, we introduce a duality between families of Calabi-Yau hypersurfaces which includes both Batyrev and Berglund-H\"ubsch-Krawitz mirror constructions. This is given in terms of a polar duality between pairs of polytopes $\Delta_1\subseteq \Delta_2$, where $\Delta_1$ and $\Delta_2^*$ are canonical.

math.AG↗

arXiv2014

Low degree hypersurfaces of projective toric varieties defined over a $C_1$ field have a rational point

Quasi algebraically closed fields, or $C_1$ fields, are defined in terms of a low degree condition. Namely, the field $K$ is $C_1$ if every degree $d$ hypersurface of the projective space $\mathbb{P}_K^n$ contains a $K$-point as soon as $d\leq n$. In this article we define a notion of low toric degree generalizing this condition for hypersurfaces of simplicial projective split toric varieties. This allows us to prove a particular case of the $C_1$ conjecture of Koll\'{a}r, Lang and Manin : any smooth separably rationally connected variety that can be embedded as such a hypersurface over a $C_1$ field has a rational point. Our results are based on the fact that the ambient toric varieties are Mori Dream Spaces : they are naturally endowed with homogeneous coordinates and their Minimal Model Program works in all cases.

math.AG↗