User dennis ochse - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:22:02Z http://mathoverflow.net/feeds/user/18305 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107972/reference-request-virtual-fundamental-class-of-moduli-of-stable-maps Reference request: virtual fundamental class of moduli of stable maps Dennis Ochse 2012-09-24T11:30:55Z 2012-09-25T00:21:26Z <p>Let <code>$f:U\longrightarrow \overline{M}_{0,n}(\mathbb{P}^m,d)$</code> is the universal family with morphism $\pi:U\longrightarrow\mathbb{P}^m$ and let $X\subset\mathbb{P}^m$ be a hypersurface defined by a global section $x$ of $\mathcal{O}(X)$ then <code>$f_{\ast}\pi^{\ast}\mathcal{O}(X)$</code> is a vectorbundle on $\overline{M}_{0,n}(\mathbb{P}^m,d)$ with global section <code>$f_{\ast}\pi^{\ast}x$</code>. It seems to be known that the zero locus of the section <code>$f_{\ast}\pi^{\ast}x$</code> is the virtual fundamental class of <code>$\overline{M}_{0,n}(X,\beta)$</code> (for a suitable $\beta$ ).</p> <p>Does anyone know a good reference for this? I had a look at the paper of Behrend and Fantechi about virtual fundamental classes, but I was not able to figure this out from that paper. Thank you in advance. </p> http://mathoverflow.net/questions/98555/chern-classes-of-vector-bundles-on-a-stack Chern classes of vector bundles on a stack Dennis Ochse 2012-06-01T09:09:57Z 2012-06-01T20:28:24Z <p>Is there literature about chern classes of vector bundles on DM-stacks? I had a look at a lot of different papers about intersection theory on stacks and related stuff and this seems to be known, but I couldn't find a good reference on this topic. The strongest statement I could find was "there are chern classes in the Chow-cohomology satisfying all the usual properties".</p> <p>A particular thing I am interested in, is the projection formula, i.e. given a proper morphism of DM-stacks $p:M\longrightarrow N$, a vectorbundle $E$ on $N$ and a cycle <code>$\alpha\in A_*(M)_\mathbb{Q}$</code> is it true that <code>$p_*(c_i(p^*E)\cap\alpha)=c_i(E)\cap p_*\alpha$</code>? I guess this is one of the usual properties, but does it hold for any proper morphism, or only representable ones? </p> <p>What I did was trying to check the definition of the Chow-cohomology from Vitsoli's paper on Intersection theory on stacks for chern classes, but I got already stuck at the compatibility with the Gysin homomorphism, because of my poor knowledge about normal bundles and stacks... so I gave up trying to prove it myself.</p> <p>Can anyone help me out with a hint to more detailed literature, or some hints how the proof of the projection formula works? I would also be happy with some precise statement under which circumstances the projection formula holds. Thank you in advance.</p> http://mathoverflow.net/questions/87248/are-there-one-dimensional-ideals-in-any-local-ring Are there one-dimensional ideals in any local ring Dennis Ochse 2012-02-01T16:21:41Z 2012-02-15T17:17:11Z <p>I would like to know if it is always possible to find a one-dimensional ideal in a local commutative ring... actually I am interested in finding a curve through a point on a scheme (locally). If the ring is of finite dimension it should be obvious, but does anybody know about the situation in more general rings? </p> http://mathoverflow.net/questions/77211/stable-reduction-for-maps Stable reduction for maps Dennis Ochse 2011-10-05T08:24:18Z 2011-10-05T09:42:59Z <p>I would like to explicitly compute the limit of a family of stable maps in $\overline{M}_{0,n}(\mathbb{P}^r,d)$. I know in principle how this works for families of curves without maps as I found a lot of literature on stable reduction for this case, like Harris and Morrison's "Moduli of Curves". Is there any literature which explains how to do this in case of stable maps? </p> <p>I don't know if it helps but I am mostly interested in families of the form $\mathbb{C}^*\times\mathbb{P}^1$ and interested in the special fibre over $0$. For example a family like $(t,(z_0:z_1))\mapsto(tz_1^3:(z_1-z_0)(z_1+z_0)^2)$ in $\overline{M}_{0,3}(\mathbb{P}^1,3)$ where the marked points are $(1:0),(1:1)$ and $(1:-1)$ in each fibre. What I tried is blowing up at the points where the map is ill-defined for $t=0$, namely $(0,(1:1))$ and $(0,(1:-1))$ but that didn't work out and I just don't see what could have gone wrong.</p> <p>I would be very grateful if anybody had suggestions what I did wrong, how to do it right or hints to the literature.</p> http://mathoverflow.net/questions/107972/reference-request-virtual-fundamental-class-of-moduli-of-stable-maps Comment by Dennis Ochse Dennis Ochse 2012-10-01T10:57:02Z 2012-10-01T10:57:02Z Thank you for the hints, they were very helpful http://mathoverflow.net/questions/107972/reference-request-virtual-fundamental-class-of-moduli-of-stable-maps Comment by Dennis Ochse Dennis Ochse 2012-09-24T11:55:48Z 2012-09-24T11:55:48Z Thank you very much, now it looks much better :) http://mathoverflow.net/questions/107972/reference-request-virtual-fundamental-class-of-moduli-of-stable-maps Comment by Dennis Ochse Dennis Ochse 2012-09-24T11:36:07Z 2012-09-24T11:36:07Z I'm sorry, but somhow it does not show the Latex-code correctly on my Browser... I hope that's only a problem of my computer :( http://mathoverflow.net/questions/68623/some-questions-on-moduli-of-stable-maps/68638#68638 Comment by Dennis Ochse Dennis Ochse 2012-09-21T08:35:50Z 2012-09-21T08:35:50Z Does anyone know a good reference for the fact in 2) that the virtual fundamental class of $\overline{M}_{0,k}(X,\beta)$ is the zero locus of that section? http://mathoverflow.net/questions/77211/stable-reduction-for-maps/77215#77215 Comment by Dennis Ochse Dennis Ochse 2011-10-05T09:25:53Z 2011-10-05T09:25:53Z Thank you, this was very helpful to at least understand the example :) Do you also know a way that is more like a recipe that also works for cases where the map is not just a cover of the projective line?