Tilting bundles and modifying modules in relative dimension one

Abstract

For projective morphisms $π:Y\to X=\operatorname{Spec}R$ over noetherian complete local rings, with fibres of dimension at most one and $\mathbf{R}π_*\mathcal{O}_Y\cong\mathcal{O}_X$, we construct a tilting fan from cones of pretilting bundles. Maximal cones parametrize tilting bundles, and facet adjacency encodes irreducible mutation. Completeness in positive rank is equivalent to finiteness of tilting bundles modulo line-bundle twists. For closed fibres that are split ADE highest-root cycles or their Stein contractions, we identify tilting bundles with chambers of restricted affine Tits cones and determine the fan. Over normal Gorenstein complete local domains, reflexive pushforward takes pretilting bundles on normal projective birational models with fibres of dimension at most one and $π^!\mathcal{O}_X\cong\mathcal{O}_Y$ to modifying modules. Pushforwards of canonical tilting bundles recover models as module blowups and detect contractions by additive containment, yielding an Auslander--McKay embedding. The hypersurfaces $R=S[[u,v]]/(uv-f)$, with $S$ a regular complete local ring and $f\in S$ a nonzero nonunit, generalize type $A$ Du Val surfaces and $cA_n$ threefolds to arbitrary dimension. We classify modifying modules by divisor flags and rank-one reflexive twists; flag blowups turn the embedding into an equivalence. If $f$ has $r\ge2$ prime factors counted with multiplicity, reflexive pushforward on any maximal-flag model gives a mutation-preserving surjection from tilting bundles, parametrized by chambers of affine type $\widetilde{A}_{r-1}$, to maximal modifying modules. Its fibres are affine Weyl group orbits determined by repeated factors. All maximal-flag models and maximal modification algebras are derived equivalent. These algebras are noncommutative crepant resolutions exactly when $S/(p)$ is regular for every prime factor $p$ of $f$.