arXiv subjects
Publications and source records attributed to Sara Mehidi.
For a morphism $f : X \to Y$ of schemes, we give a tropical criterion for which points of $Y$ (valued in a field, discrete valuation ring, number ring, or Dedekind domain) lift to $X$. Our criterion extends the firmaments of Abramovich to a wide range of morphisms, even logarithmic stable maps.
We generalize the logarithmic purity theorem of Fujiwara-Kato-Mochizuki to torsors which arise in the Kummer log flat topology under finite flat linearly reductive group schemes. We then give a stack-theoretic interpretation of our purity theorem via root stacks, relating it to the valuative criterion of properness for tame algebraic stacks of Bresciani-Vistoli. Finally, we construct a logarithmic Nori fundamental group scheme of a log regular log scheme classifying such torsors, and compare it with the classical Nori fundamental group and the tame fundamental group.
We prove that smooth non-klt toric orbifolds are separably Campana rationally connected, extending the result in the klt case. We also show that there always exists a positive characteristic in which a singular weighted projective space, viewed as a non-klt Campana orbifold, is not separably Campana rationally connected.
We show that log flat torsors over a family $X/S$ of nodal curves under a finite flat commutative group scheme $G/S$ are classified by maps from the Cartier dual of $G$ to the log Jacobian of $X$. We deduce that fppf torsors on the smooth fibres of $X/S$ can be extended to global log flat torsors under some regularity hypotheses.
Let $R$ be a discrete valuation ring of field of fractions $K$ and of residue field $k$ of characteristic $p > 0$. In an earlier work, we studied the question of extending torsors on $K$-curves into torsors over $R$-regular models of the curves in the case when the structural $K$-group scheme of the torsor admits a finite flat model over $R$. In this paper, we first give a simpler description of the problem in the case where the curve is semistable. Secondly, if $R$ is assumed to be Henselian and Japanese, we solve the problem of extending torsors even if the structural group does not admit a finite flat $R$-model.
Let $R$ be a discrete valuation ring with field of fractions $K$ and residue field $k$ of characteristic $p>0$. Given a finite commutative group scheme $G$ over $K$ and a smooth projective curve $C$ over $K$ with a rational point, we study the extension of pointed fppf $G$-torsors over $C$ to pointed torsors over some $R$-regular model $\mathcal{C}$ of $C$. We first study this problem in the category of log schemes: given a finite flat $R$-group scheme $\mathcal{G}$, we prove that the data of a pointed $\mathcal{G}$-log torsor over $\mathcal{C}$ is equivalent to that of a morphism $\mathcal{G}^D \to \mathrm{Pic}^{log}_{\mathcal{C}/R}$, where $\mathcal{G}^D$ is the Cartier dual of $\mathcal{G}$ and $\mathrm{Pic}^{log}_{\mathcal{C}/R}$ the log Picard functor. Then, we deduce a criterion for the extension of torsors: it suffices to find a finite flat model of $G$ over $R$ for which a certain group scheme morphism to the Jacobian $J$ of $C$ extends to the Néron model of $J$. In this context, we compute the obstruction for the extended log torsor to come from an fppf one. In a second part, we generalize a result of Chiodo which gives a criterion for the $r$-torsion subgroup of the Néron model of $J$ to be a finite flat group scheme, and we combine it with the results of the first part. Finally, we give two detailed examples of extension of torsors when $C$ is a hyperelliptic curve defined over $\mathbb{Q}$, which will illustrates our techniques.