arXiv subjects
Publications and source records attributed to Morgan V Brown.
We demonstrate the construction of singular log Calabi-Yau $4$-folds such that the dual complex of the boundary is homeomorphic to a Lens space from a log Calabi-Yau surface with action of a finite cyclic group. We explicitly obtain the Lens spaces $L(3,1)$, $L(5,1)$, and $L(5,2)$ in this way.
Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $Δ$ of a log canonical pair $(X,Δ)$, and also appear as limits of canonically polarized varieties in moduli theory. For certain three-fold pairs $(X,Δ)$ we show how to compute the PL homeomorphism type of the dual complex of a dlt minimal model directly from the normalization data of $Δ$.
A recent paper of Totaro develops a theory of $q$-ample bundles in characteristic 0. Specifically, a line bundle $L$ on $X$ is $q$-ample if for every coherent sheaf $\mathcal{F}$ on $X$, there exists an integer $m_0$ such that $m\geq m_0$ implies $H^i(X,\mathcal{F}\otimes \mathcal{O}(mL))=0$ for $i>q$. We show that a line bundle $L$ on a complex projective scheme $X$ is $q$-ample if and only if the restriction of $L$ to its augmented base locus is $q$-ample. In particular, when $X$ is a variety and $L$ is big but fails to be $q$-ample, then there exists a codimension 1 subscheme $D$ of $X$ such that the restriction of $L$ to $D$ is not $q$-ample.