Related papers: A tour of stable reduction with applications
We study families of rational curves on an algebraic variety satisfying incidence conditions. We prove an analogue of bend-and-break: that is, we show that under suitable conditions, such a family must contain reducibles. In the case of…
We give a new proof of the Semistable Reduction Theorem for curves. The main idea is to present a curve $Y$ over a local field $K$ as a finite cover of the projective line $X=\PP^1_K$. By successive blowups (and after replacing $K$ by a…
We construct a smooth Deligne-Mumford compactification for the moduli space of curves with an m-tuple of spin structures using line bundles on quasi-stable curves as limiting objects, as opposed to line bundles on stacky curves. For all m,…
We study asymptotic properties of the modular representation theory of symmetric groups and investigate modular analogs of stabilization phenomena in characteristic zero. The main results are equivalences of categories between certain…
We constructed a projective moduli space of semistable torsion free sheaves with `fixed determinant' on a reducible curve. When a family of smooth curves degenerates to the reducible curve, our moduli space is a degeneration of the moduli…
We define a Deligne-Mumford stack X_{D,r} which depends on a scheme X, an effective Cartier divisor D\subset X, and a positive integer r. Then we show that the Abramovich-Vistoli moduli stack of stable maps into X_{D,r} provides…
We introduce a class of complex surface singularities - the blow-$ADE$ singularities - which are likely to be stable with respect to $\mu^*$-constant deformations. We prove such a stability property in several special cases. Here, we…
We discuss how the motivic integration will be generalized to wild Deligne-Mumford stacks, that is, stabilizers may have order divisible by the characteristic of the base or residue field. We pose several conjectures on this topic. We also…
We prove a semistable reduction theorem for principal bundles on curves in almost arbitrary characteristics. For exceptional groups we need some small explicit restrictions on the characteristic.
We study stability and bifurcations in holomorphic families of polynomial automorphisms of C^2. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines…
We present a symmetry result regarding stationary solutions of the 2D Euler equations in a disk. We prove that in a disk, a steady flow with only one stagnation point and tangential boundary conditions is a circular flow, which confirms a…
We introduce a set of combinatorial techniques for studying the simplicial bounded cohomology of semi-simplicial sets, simplicial complexes and posets. We apply these methods to prove several new bounded acyclicity results for…
We study ample stable vector bundles on minimal rational surfaces. We give a complete classification of those moduli spaces for which the general stable bundle is both ample and globally generated. We also prove that if $V$ is any stable…
We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation…
As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-Macaulay modules over a Gorenstein local algebra. We shall give several necessary and/or sufficient conditions for the stable degeneration.…
We show that a toroidal morphism can be reduced to a weakly semistable one in a universal way if we allow families to be modified to Deligne-Mumford stacks instead of schemes.
We develop foundational aspects of stability theory in affine logic. On the one hand, we prove appropriate affine versions of many classical results, including definability of types, existence of non-forking extensions, and other…
We generalize the notion of expanded degenerations and pairs for a simple degeneration or smooth pair to the case of smooth Deligne-Mumford stacks. We then define stable quotients on the classifying stacks of expanded degenerations and…
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the…
We introduce the notion of Mukai regularity (M-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and…