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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
4
votes
0
answers
198
views
How can I describe the monodromy of this variation of singular curves?
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 …
3
votes
1
answer
281
views
Is there a notion of injective, projective, flat, dimension for a differential graded algebra?
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 …
4
votes
1
answer
261
views
Are there algorithmic tools for computing poincare residues?
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
$$
\frac{dw}{w^k}\wedge …
4
votes
0
answers
226
views
Is there an analogue to the koszul complex for constructible sheaves?
Given a variety $X$ and a complete-intersection morphism
$$
Y \to X
$$
is there an analogue of the Koszul complex for $\mathcal{O}_Y \in \textbf{Coh}(X)$ in the setting of constructible sheaves? Meani …
4
votes
0
answers
341
views
Does hypercohomology of the Koszul complex compute sheaf cohomology?
Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, meaning $ …
-1
votes
1
answer
464
views
How do I find the algebra representing the projective bundle of a direct sum of line bundles...
I am trying to learn how to compute the projective bundle $\mathbb{P}(\mathcal{O}(a_1)\oplus \cdots \mathcal{O}(a_k))$ over some projective space using relative proj. How can I find a presentation for …
1
vote
Examples for Decomposition Theorem
A nice example of the decomposition theorem for a proper morphism of smooth varieties is the blowup of the points $p_i$ lying at the intersection of two plane curves; that is, let $f, g \in \mathbb{C} …
2
votes
0
answers
226
views
What is the motivation behind the definition for a smooth differential graded category?
Let $\mathcal{A}$ be an $\mathbb{F}$-linear differential graded category. It is said to be smooth if it is a perfect complex over the differential graded category $\mathcal{A}^\circ\otimes_\mathbb{F}\ …
4
votes
What are examples of D-modules that I should have in mind while learning the theory?
I will edit this question as I learn more, but here's one useful example: Consider the $\mathcal{D}_{\mathbb{A}^1}$-module
$$
\frac{\mathcal{D}_{\mathbb{A}^1}}{\mathcal{D}_{\mathbb{A}^1}(t\partial_t - …
17
votes
2
answers
1k
views
What is the motivation behind the characteristic variety of a D-module and what does it's ge...
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
$$
\text{Char}(\mathcal{M}):= V\left(\sqrt{Ann(\mathcal{M …
0
votes
1
answer
322
views
How can I show flatness for projective morphisms?
Are there any homological checks I can use to check if a projective morphism is flat? For example, I would expect the following projective morphism to be flat
$$
\textbf{Proj}\left( \frac{\mathbb{C}[s …
6
votes
1
answer
740
views
What tools can I use to compute the cohomology of the fibers of a Lefschetz Pencil?
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 …
2
votes
1
answer
320
views
Where can I find a proof of identity of $H^1(X,T_X)$ and a quotient by the jacobian?
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 …
23
votes
2
answers
2k
views
What are examples of D-modules that I should have in mind while learning the theory?
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 …
7
votes
0
answers
279
views
Is there a derived geometric interpretation of morse functions?
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 …