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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

**2**

votes

**0**answers

In SGA 1 Expose XIII, $\S2.1$, they give a definition of a "normal crossings divisor relative to $S$" which is a bit difficult to parse (and has some typos).
My first question is: Is the following a …

asked Aug 1 '14 by Will Chen

**3**

votes

**0**answers

Let $X\rightarrow S$ be an arithmetic surface (this means, in the language of Qing Liu's book, $X$ is a regular integral scheme of dimension 2, projective and flat over a Dedekind scheme $S$ of dimens …

asked Apr 28 '17 by Will Chen

**5**

votes

**0**answers

Let $f : \mathcal{X}\rightarrow\mathcal{Y}$ be a morphism of algebraic stacks which is a gerbe in the sense of the stacks project. Must $f$ induce a surjection of etale fundamental groups? I'm happy t …

asked Dec 17 '20 by Will Chen

**1**

vote

**1**answer

In section 5.5 of Deligne + Mumford's paper "Irreducibility of the Moduli Space of Curves", they introduce the notion of Teichmuller level structures.
You can find the paper here: http://publications …

asked Jul 9 '14 by Will Chen

**3**

votes

**1**answer

Let $X,Y$ be finite etale $T$ schemes for some scheme $T$ (assume the maps $X\rightarrow T,Y\rightarrow T$ are surjective). Then the sheaf $\mathcal{Hom}_T(X,Y)$ on $\text{Sch}/T$ (with the etale topo …

asked Aug 7 '14 by Will Chen

**2**

votes

**1**answer

Let $R$ be a strictly henselian local ring of dimension 2, satisfying Serre's condition $S_2$. Let $X = \text{Spec }R$, and let $f : Y\rightarrow X$ be a finite morphism inducing an isomorphism over t …

asked Jun 1 '17 by Will Chen

**5**

votes

**2**answers

For a scheme $X$, let $LE(X)$ denote the lisse-etale site on $X$. This is the full subcategory of $\textbf{Sch}/X$ consisting of smooth morphisms to $X$, equipped with the etale topology. Let $\mathca …

asked Nov 2 '17 by Will Chen

**2**

votes

I believe that this is true - that is, "$p$ is finite flat totally ramified with ramification index equal to the order of the group of automorphisms of $X_0$ modulo the subgroup of those which extend …

answered Nov 30 '17 by Will Chen

**5**

votes

As Piotr pointed out, the answer to your question is no as stated.
I just wanted to add a further comment (too long for an actual comment)
I guess what you're thinking is that it suffices to check e …

answered Jun 22 '18 by Will Chen

**4**

votes

**0**answers

What are some explicit examples (e.g., by explicitly describing its Hopf algebra) of finite unramified group schemes? (Ie, the sort of group schemes which appear as automorphism groups of objects para …

asked Jun 11 '17 by Will Chen

**1**

vote

Thanks to Daniel Bergh's answer, after an hour of thinking about this, I finally understand the problem.
Fundamentally, the confusion stemmed from a bad intuition about inductive limits (even of sets …

answered Nov 7 '17 by Will Chen

**3**

votes

I can only address your question (1), to which I believe the answer is yes.
Your ring $R$ is a complete intersection local ring of dimension $n$, and hence if $n\ge 3$, then by Stacks 0BPD, it satisf …

answered Apr 10 '18 by Will Chen

**3**

votes

**1**answer

Let $X_K$ be a smooth proper curve over a field $K$, and let $S$ be a Dedekind scheme with function field $K$.
Let $X$ be the proper regular minimal model of $X_K$ over $S$.
Let $Y_K$ be another smo …

asked Sep 30 '16 by Will Chen

**5**

votes

**1**answer

Is there a standard reference on stacks which discusses (relative) normalization?
This older question seemed to link to someone's notes, but the link is now broken. In any case, it would be nice to h …

asked Jun 8 '17 by Will Chen

**5**

votes

**1**answer

Let $\mathcal{M}$ be a smooth 1-dimensional Deligne-Mumford stack with finite diagonal, and let $M$ be its coarse moduli scheme (all over some field $k$). Suppose $M$ is also smooth over $k$, and that …

asked Jul 11 '17 by Will Chen