For a fixed smooth curve C and reductive group G, we prove the connectedness of the fibers of the Hitchin fibration with source the stack of G-Higgs bundles of fixed degree (no semistability). To our knowledge, the strategies in the literature for proving connectedness of the fibers of the Hitchin fibration don't apply to the whole stack (which is not of finite type). Instead, we use some Gm equivariant geometry and resolution of surfaces.

This note is based on a question by user Math Display on Mathoverflow. We show that torsors over a (not necessarily reductive) affine algebraic group can be characterized in terms of a weak notion of geometric quotient.

We prove a structural result about the connected components of the relative Picard scheme of a smooth proper scheme with geometrically irreducible fibers over a strictly Henselian local ring in characteristic 0.

In this note we observe that the global topology of morphisms that are equivariant with respect to actions of the multiplicative group is sometimes controlled by the fiber over the fixed locus of the target.

This a short note written by Siqing Zhang explaining how Cartier descent is a special case of fppf descent. Thanks to Siqing for letting me include this note here.

This note is written joint with Mark Andrea de Cataldo. We show that, under mild hypotheses, any finite (schematic) morphism of reduced stacks can be canonically factored as the composition of a finite radicial morphism followed by a finite generically etale morphism.

This note contains a proof of a version of Hartogs's theorem for maps into the stack BG, where G is a smooth affine geometrically reductive groups scheme (not necessarily locally embeddable).

In this note we prove two results about the residual gerbes. The first one is that the residual gerbe of a point in a quasiseparated algebraic stack is the limit of all locally closed substacks containing that point. The second is that the topological space of the base-change of the inclusion of a residual gerbe of a point in a quasiseparated stack always acquires the subspace topology. We provide an example to show that this is not true if we remove the quasiseparated hypothesis.

This is a short note written by Andres Ibanez Nunez. It sketches a proof of the well-know fact that the tangent space of a stack at a closed point coincides with the normal bundle (given that certain hypotheses are satisfied). Thanks to Andres for letting me include this here.

This is a letter to Andres Ibanez Nunez on November 28 2022. It sketches a proof of flatness of the kernel of a morphism with source a product of copies of multiplicative groups, under some mild assumptions. It's a small direct proof just for fun.

This is a letter to Gyujin Oh written on October 5 2022. It oulines a proof that the formation of the moduli space of connections for a smooth family of curves commutes with base-change on the base of the family, as long as we invert enough primes.

In this note I record my favorite example of a faithfully flat morphism that is not fpqc. It illustrates why we can't expect to be able to check certain properties of morphisms (e.g. being an isomorphism) after base-change via any faithfully flat morphisms.

This is a letter to Andres Ibanez Nunez written on September 2 2022. It contains a direct proof that a good moduli space morphism between two gerbes is itself a gerbe, as long as the target has affine diagonal.

We give a local criterion for smoothness of a morphism between smooth schemes in terms of their cotangent spaces. The proof is a small diagram chase argument using the cotangent complex. Corollary 3 in this note can be used to prove the converse in the main proposition in this post in positive characteristic, thus answering a question of Sean Cotner.

A list of useful criteria for flatness.

This contains a rough sketch of a proof that the formation of scheme theoretic image for a (not necessarily quasicompact) morphism commutes with flat base change, under the assumption that the target is quasicompact, quasiseparated, and essentially free over the base. We use the concrete description of coherator for sheaves on a quasicompact quasiseparated scheme.

Notes on canonical/bar resolutions, with examples.

Some expository notes giving intuition on the notion of equivariant sheaf.