11
votes
Relation between motives and geometric Langlands
First note that the formulation of geometric Langlands (the eigensheaf part, not the full conjectured equivalence of categories) is that associated to a local system on $X$ there exists an eigensheaf ...
6
votes
Zero set of prime ideal
Let $k = \mathbb{R}$ and let $P$ be the prime ideal of $\mathbb{R}[x,y]$ generated by $h := y^4 + y^2 + (x^2-1)^2$. According to Sage¹ (though I'm sure this is easy to check by other means), $h$ is ...
5
votes
Reference request: good reduction equivalent to crystalline étale cohomology
As Satan's Minion says, the good reduction case is
R. Coleman, A. Iovita, The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J. 97 (1999), 171--215.
For the semistable ...
3
votes
Reconciling the affine grassmannian and the based loop group
There are many spaces of maps $S^1\to K$ (hence many version of the affine Grassmannian) one might want to consider.
Let's list them:
• Algebraic maps (i.e. algebraic maps from $\mathbb C^\times$ to $...
3
votes
Accepted
Lifting of quadrics containing a curve
1-normality is sufficient, i.e. it suffices that $H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(1))\to H^0(C, \mathcal O_C(1))$ is surjective.
Indeed, choose class $q \in H^0(\mathbb P^r, \mathcal O_{\...
3
votes
Accepted
Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)
I'm going to use $g$ for the equation of the curve since using $f$ for both the equation of the curve and the map to $\operatorname{Spec}(\mathbb C[s,t])$ leads to ambiguity.
The situation just using ...
3
votes
What is the status of the theory of motives?
This is not very recent but it is worth mentioning. One approach to the existence of category of mixed motives $MM(k)$ ($k$ a field of characteristic zero) is via the existence of the motivic Galois ...
2
votes
Accepted
Residues and blow ups
First, I want to clarify the concept of the residue that you're thinking of to algebraic geometers: Start with a complex manifold $X$, and two smooth divisors $D_1$ and $D_2$ meeting transversely. ...
2
votes
One question about K-moduli space of smooth plane conic curves
There are two ways to see that $X$ is isomorphic to $\mathbb{P}(1,1,4)$, both using the fact that $X$ isotrivially degenerates to $\mathbb{P}(1,1,4)$.
(1) By Hacking's paper https://arxiv.org/pdf/math/...
2
votes
Why is this polynomial factorizable?
Here is an idea that should lead to a human way to verify this.
Take a triple $(a_1,a_2,a_3)$ for which $a_1+a_2+a_3=0$. For example, $a_1=a=-a_2$, and $a_3=0$ or $a_i=a\cdot\omega^i$ where $\omega^3=...
1
vote
Accepted
Shrinking the base field of an affine variety
A standard reference is Chapter 6 of "Neron Models" of Bosch, Lütkebohmert and Raynaud, Springer, 1990. The original Grothendieck's exposés in the Cartan Seminar (1960--1961) are available ...
1
vote
Riemann-Hilbert problem via quiver description
I don't think the map you are considering is surjective in general. Let us consider the case where where you have $m=4$ points and all the $A_i$ lie in the conjugacy class of
$$\begin{pmatrix} 0 & ...
1
vote
References for $K$-orbits in $G/B$
For what its worth the Atlas of Lie Groups and Representation software https://www.liegroups.org computes the K orbits on G/B for any (connected complex reductive) G and any (algebraic) involution. ...
1
vote
Application of higher categories in algebra
There are tons of such applications. For example, less than two years ago, they were used to prove the Redshift Conjecture, a statement about iterated algebraic $K$-theory. I previously wrote an ...
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