User nullstellensatz - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:15:13Z http://mathoverflow.net/feeds/user/30850 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120699/on-the-local-structure-of-stacks On the local structure of stacks Nullstellensatz 2013-02-03T20:09:18Z 2013-03-06T09:46:33Z <p>1) Is it true that any Deligne-Mumford stack is locally a quotient stack $[X/G]$ with a finite group $G$?</p> <p>2) Is it true that any Deligne-Mumford stack can be globally presented as a quotient stack $[X/G]$ with a non necessarily finite group $G$? For example, Geigle and Lenzing give such a presentation for stacky projective lines <a href="http://www.math.washington.edu/~smith/WPLSeminar/GL87.pdf" rel="nofollow">here</a>.</p> <p>3) What about Artin stacks?</p> http://mathoverflow.net/questions/120567/on-the-coarse-moduli-space-of-a-stack On the coarse moduli space of a stack Nullstellensatz 2013-02-02T01:08:15Z 2013-02-04T08:04:14Z <p>Consider a stack $\mathcal{X}$ over $\mathbb{C}$ as a category fibred in groupoids over the category of schemes. Let $\mathcal{X}^s$ be the $\pi_0$ of this category, i.e. objects of $\mathcal{X}^s$ are the objects in $\mathcal{X}$ and morphisms of $\mathcal{X}^s$ are the morphisms in $\mathcal{X}$ modulo automorphisms of objects. It "kills" the groupoid structure, so I think it is possible to consider $\mathcal{X}^s$ as a category fibred in sets over the category of schemes. Assume $\mathcal{X}^s$ is represented by a scheme. Should it be the coarse moduli space for $\mathcal{X}$?</p> http://mathoverflow.net/questions/119647/on-the-flatness-of-certain-morphism On the flatness of certain morphism Nullstellensatz 2013-01-23T11:26:00Z 2013-01-24T22:53:14Z <p>Assume $X, Z, S$ are shemes of finite type over $\mathbb{C}$, $X$ is also irreducible and reduced, $\phi: Z\to S$ is affine flat morphism with reduced connected fibers, $\psi: Z\to X$ is such that $\phi \times \psi$ is finite. Is $\text{Im}(\phi \times \psi)\to S$ flat? Maybe, under some additional conditions?</p> <p>For example, let $S=\mathbb{A}^1$, $Z=X=\mathbb{A}^2$, $\phi$ is given by $(x,y)\mapsto x$, $\psi$ is given by $(x,y)\mapsto (xy, y^2)$. The morphism $\phi\times\psi$ is bijective outside the fiber $Z_0$, where it is 2 to 1. However, $\text{Im}(\phi \times \psi)\to S$ is flat.</p> <p>Thanks.</p> http://mathoverflow.net/questions/120699/on-the-local-structure-of-stacks Comment by Nullstellensatz Nullstellensatz 2013-02-04T16:41:34Z 2013-02-04T16:41:34Z @pranavk: I deliberately did not specify, because I am interested in any kind of results. http://mathoverflow.net/questions/120567/on-the-coarse-moduli-space-of-a-stack/120603#120603 Comment by Nullstellensatz Nullstellensatz 2013-02-03T14:00:06Z 2013-02-03T14:00:06Z And is it right that the sheafification of $\mathcal{X}^s$ is a scheme in the case of the quotient stack $[\mathbb{A}^1/\mathbb{Z}_n]$? http://mathoverflow.net/questions/120567/on-the-coarse-moduli-space-of-a-stack/120603#120603 Comment by Nullstellensatz Nullstellensatz 2013-02-03T13:50:43Z 2013-02-03T13:50:43Z Thanks for the detailed answer, Dan. So, do I understand correctly that, if the sheafification of $\mathcal{X}^s$ is represented by a scheme, then it is a coarse moduli for $\mathcal{X}$? http://mathoverflow.net/questions/120567/on-the-coarse-moduli-space-of-a-stack/120603#120603 Comment by Nullstellensatz Nullstellensatz 2013-02-02T17:10:36Z 2013-02-02T17:10:36Z And what if $\mathcal{X}$ is the classifying stack BG or, for example, the quotient stack $[\mathbb{A}/\mathbb{Z}_n]$? Isn't $\mathcal{X}^s$ the coarse moduli for $\mathcal{X}$ in these cases? http://mathoverflow.net/questions/120459/oarse-moduli-space-of-a-stack Comment by Nullstellensatz Nullstellensatz 2013-02-01T18:55:28Z 2013-02-01T18:55:28Z Objects in $M(S)$ could posses non-trivial automorphisms, so $M(S)\to M′(S)$ &quot;kills automorphisms&quot;, i.e. sends isomorphic objects in $M(S)$ to the same object in $M′(S)$. I'm absolutely sure that in my situation $M$ can't be representing by a scheme, but maybe I am saying something wrong (I'm begginer in stacks)... http://mathoverflow.net/questions/120459/oarse-moduli-space-of-a-stack Comment by Nullstellensatz Nullstellensatz 2013-02-01T18:26:16Z 2013-02-01T18:26:16Z @Martin: Ok, I made some addition. http://mathoverflow.net/questions/120459/oarse-moduli-space-of-a-stack Comment by Nullstellensatz Nullstellensatz 2013-02-01T18:26:11Z 2013-02-01T18:26:11Z @David: Is it sufficient condition? I don't think so... Also I think that this is not applicable for my situation... http://mathoverflow.net/questions/119572/on-the-closedness-of-the-image-and-the-flatness-of-certain-morphisms/119587#119587 Comment by Nullstellensatz Nullstellensatz 2013-01-22T19:12:44Z 2013-01-22T19:12:44Z I'm more interested in the case, when $\phi$ has connected fibers. Is there a contrexample to the flatness of $\text{Im}(\phi\times\psi)\to S$ in this case? http://mathoverflow.net/questions/119572/on-the-closedness-of-the-image-and-the-flatness-of-certain-morphisms/119587#119587 Comment by Nullstellensatz Nullstellensatz 2013-01-22T18:33:05Z 2013-01-22T18:33:05Z Thank you, S&#225;ndor. But could there be some additional conditions on morphisms and schemes, under witch 2 will be true? I think, for my purposes it can be assumed that $\phi$ is affine morphism with connected reduced fibers and X is an affine variety. Will $\text{Im}(\phi\times\psi)\to S$ be flat then?