Jesko Hüttenhain
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 Dec 21 revised Can every variety appear as singular locus? LaTeX Dec 20 comment How to describe the points of a quotient stack? The image of a $G$-equivariant map from $G$ is certainly always an orbit. Let $\alpha:G\to X$ be such a map, let $x:=\alpha(1)$ and then you have $\alpha(g)=\alpha(g\cdot 1)=g.\alpha(1)=g.x$ for all $g\in G$. Dec 20 comment When does “closed immersion” pass to the quotient? If $\dim(Y)=\dim(X)-q$ and $\dim(H)=\dim(G)-p$ then there will be examples where $\dim(Y/H)=\dim(X)-\dim(G)+p-q$ and $\dim(X/G)=\dim(X)-\dim(G)$ (and everything remains a variety). So, this feels to me like you'd need at least $q>p$ to make this work. However, even much more pathological things can happen. Dec 19 comment When does “closed immersion” pass to the quotient? Unlikely to be true if formulated this way. Choose $H=\{1\}$, then your morphism $\bar f\colon Y\to X/G$, dimension alert: Choose $Y=X$ and $G$ acting transitively on $X$, then you'd have $\bar f: X\to\{\ast\}$ which will in general not be injective. Dec 17 comment Can every variety appear as singular locus? If $V$ is smooth, the answer is easily yes: Let $C$ be a curve with a unique singular point $p$. Then, the singular locus of $C\times V$ is equal to $\{p\}\times V\cong V$. It's not that easy when $V$ itself is singular, though. Dec 5 comment Restriction of closed immersion to closed subset is a closed immersion Since the inclusion $\iota: Z\to X$ is a closed immersion, it induces a surjection $\mathcal{O}_{X,\iota(z)}\twoheadrightarrow\mathcal{O}_{Z,z}$ for a point $z\in Z$. You furthermore know that $f$ induces a surjection $\mathcal O_{Y,f(z)}\twoheadrightarrow\mathcal O_{X,z}$ for any $z\in X$, in particular for $z=\iota(z)\in Z$. The composition of these two surjections is the local map induced by $f\circ\iota$, and it is a surjection as well. Does this answer the question? Dec 3 awarded Popular Question Nov 12 answered Why is the secant variety a variety? Nov 12 comment Why is the secant variety a variety? @kaiser: I think Hoot's question is relevant. Is a variety always irreducible by your definition? It is an unfortunate communication barrier in algebraic geometry that "variety" is not entirely unambiguous. Nov 12 comment Why is the secant variety a variety? It is obvious that this is a variety because you take the closure. By definition, the result is a Zariski closed subset of projective space. Nov 3 answered Categorical Quotient Oct 21 answered Does the character with the following properties exist? Oct 8 answered Constructing realizations of Hilberts weak Nullstellensatz? Oct 5 comment Algebra of invariants finitely generated Definitely true if $k$ is of characteristic zero. A reference would be 27.4 in the book Lie algebras and algebraic groups by Tauvel & Yu. Oct 5 comment Prove of Sheafication of sheaf is a sheaf isomorphism In other words, what is your definition of sheafification? Also, to be on the safe side, how do you define presheaf and sheaf? =) Sep 24 comment Euler character of etale finite cover @poorna: sounds like a tame situation, but I don't know. Euler characteristic is sensitive to removing a finite set of points, so I am unsure. I think it might be best to ask this as a separate question, I am sure someone will know better. Sep 23 comment Euler character of etale finite cover @poorna: Ok, I was a bit slow there. But assuming that $\pi$ is etale outside a finite set of points does not even guarantee that the morphism is finite, it could for instance be the blow-up of a singular surface. I suspect one might be able to construct counterexamples this way. Sep 23 comment Euler character of etale finite cover @poorna: We need Hirzebruch-Riemann Roch only for the Bundle $\mathcal O_{\tilde X}$ on the smooth variety $\tilde X$, which works. The Lemma requires the morphism $\pi:\tilde X\to X$ to be flat and finite, according to Fulton. I would assume that this condition can not be left out. Flatness is implicit if both $X$ and $\tilde X$ are regular. So, I think it works if you can show that $\pi$ is flat. I do not know off the top of my hat if your conditions imply that, though. Flatness is quite hard to grasp for me. If something comes to mind, I will post it. Sep 23 revised Euler character of etale finite cover added 18 characters in body Aug 30 awarded Self-Learner