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When we have a variety and a resolution of singularities, but it is not semi-small (i.e. the dimensions of the fibres do not satisfy the right conditions), then what can we say about the intersection …

For an affine variety, I know how to compute the set of singular points by simply looking at the points where the Jacobian matrix for the set of defining equations has too small a rank. But what is …

I'm having trouble finding the closure in the definition of a blow-up. For example, take the following example, with a node at $(0,0)$ (and at some other points) (it's not a homework question, just a …

Motivation: I was reading through Frenkel's article on geometric Langlands program, and the external tensor product of two perverse sheaves occurred in the definition of the geometric Langlands conjec …

I have a basic question about the definition of $D_X$-algebras and schemes, which are defined in [BD2], Chiral algebras. I have some understanding of connections on a vector bundle, but I am not sure …

Let $A \rightarrow B$ and $C \rightarrow B$ be two maps of schemes. How can I compute the derived fiber product $A \otimes^L_B C$? I'm guessing this is a dg-scheme. For instance - let $B=\mathbb{A}^1 …

As I understand, when we have a nilpotent cone, or a nullcone of a Lie group representation, what seems to be done in a lot of the literature (e.g. Achar&Henderson-"Orbit closures in the enhanced nilp …

I was wondering what good references there are for equivariant cohomology. Specifically, I am working through the computation of $\text{H}^{\bullet}_{G(\mathcal{O})}(\text{Gr}_G)$ in http://arxiv.or …

My question is regarding the definition of $\mathcal{D}$-modules of level $m$ given in this paper. As an example, let $X=\mathbb{A}^1$ over $S=\text{Spec }\overline{\mathbb{F}_p}$; I was told that a $ …

This question is related to my other question. Consider a scheme $X$ over $S=\text{Spec}(\mathbb{k})$ where $\mathbb{k}=\overline{\mathbb{F}_p})$; let $F: X \rightarrow X$ be the Frobenius $p$-th powe …

I have a couple questions regarding the proof of Proposition $3$ (see page $10-11$ of arxiv.org/abs/math/0102039) in Bezrukavnikov's paper "Quasi-exceptional sets and equivariant coherent sheaves on t …

My question is about the paper "Affine tangles and irreducible exotic sheaves" (arxiv.org/abs/0802.1070). Background: Let $\mathcal{B}, \mathcal{N}$ denote the flag variety and nilpotent cone of $G= …

For $G = GL_n$, it is known that the generic fibers of the Hitchin fibration are the Picard stacks of line bundles on the corresponding spectral curves and the duality of Hitchin fibrations in this ca …

Here is a result whose proof uses Fourier-Mukai duality: Consider a family of abelian varieties $A \rightarrow X$, its dual $\check{A} \rightarrow X$, and a torsor $\mathcal{T}$ (for $A \rightarrow X …

Let $G$ be a simple algebraic group, $\mathcal{O}=\mathbb{C}[[t]], \mathcal{K}=\mathbb{C}((t))$ and let $\text{Gr}_G=G(\mathcal{K})/G(\mathcal{O})$ be the affine Grassmanian. My main question: Why …