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Consider the family of singular hyperelliptic curves $$ y^2 - x(x-1)^2(x-2)(x-3)(x-4)(x-t) $$ over $\mathbb{A}^1_t$. Over a generic point the fiber is a genus three curve where one of the genera comes …

Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of differe …

In Schnell's note on Computing Picard-Fuchs Equations he gives a recursive method for computing residues on hypersurfaces. In short, if you have a meromorphic differential form $$ \frac{dw}{w^k}\wedge …

Given a variety $X$ and a complete-intersection morphism $$ Y \to X $$ is there an analogue of the Koszul complex for $\mathcal{O}_Y \in \textbf{Coh}(X)$ in the setting of constructible sheaves? Meani …

Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, meaning $ …

I am trying to learn how to compute the projective bundle $\mathbb{P}(\mathcal{O}(a_1)\oplus \cdots \mathcal{O}(a_k))$ over some projective space using relative proj. How can I find a presentation for …

A nice example of the decomposition theorem for a proper morphism of smooth varieties is the blowup of the points $p_i$ lying at the intersection of two plane curves; that is, let $f, g \in \mathbb{C} …

Let $\mathcal{A}$ be an $\mathbb{F}$-linear differential graded category. It is said to be smooth if it is a perfect complex over the differential graded category $\mathcal{A}^\circ\otimes_\mathbb{F}\ …

I will edit this question as I learn more, but here's one useful example: Consider the $\mathcal{D}_{\mathbb{A}^1}$-module $$ \frac{\mathcal{D}_{\mathbb{A}^1}}{\mathcal{D}_{\mathbb{A}^1}(t\partial_t - …

Given a smooth algebraic variety $X$, and an $\mathcal{M}\in \text{Mod}(D_X)$, there is the characteristic variety of $\mathcal{M}$ defined as $$ \text{Char}(\mathcal{M}):= V\left(\sqrt{Ann(\mathcal{M …

Are there any homological checks I can use to check if a projective morphism is flat? For example, I would expect the following projective morphism to be flat $$ \textbf{Proj}\left( \frac{\mathbb{C}[s …

I'm learning about Lefschetz pencils and vanishing cycles and have looked at a few sources: http://www.math.purdue.edu/~dvb/preprints/sheaves.pdf http://www3.nd.edu/~lnicolae/Morse2nd.pdf Voisin's C …

I'm reading some notes on hodge theory by Charles Siegel which makes a claim on page 16 relating the space of deformations of a smooth projective hypersurface $X$ with the jacobian ideal. More specifi …

I've been reading about D-modules this summer in preparation for a learning seminar on intersection cohomology. Unfortunately, many of the ideas are not sticking while I learn about the theory. What a …

Given a smooth affine scheme $X = \mathbb{V}(g)$ over a field of characteristic 0, let $f:X \to \mathbb{A}^1$ be a morphism of schemes. Then, the critical locus is given by $\pi_*(dg \cap df)$ for $\p …