Deformations of pointed stable maps with "curve held rigid" or "preserving the dual graph" - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:51:37Z http://mathoverflow.net/feeds/question/99298 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99298/deformations-of-pointed-stable-maps-with-curve-held-rigid-or-preserving-the-du Deformations of pointed stable maps with "curve held rigid" or "preserving the dual graph" finite 2012-06-11T13:40:58Z 2012-06-12T14:43:06Z <p>I am reading the "<a href="http://arxiv.org/abs/alg-geom/9608011" rel="nofollow">Notes on stable maps and quantum cohomology</a>" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order deformations of a pointed stable map $\mu = (C, p_i, \mu: C \to X)$. For $C = \mathbb P^1$ they write down an exact sequence $$0 \to H^0(C,T_C) \to Def_R(\mu) \to Def(\mu) \to 0,$$ where $Def_R(\mu)$ is the space of first order deformations of $\mu$ with $C$ held rigid.</p> <blockquote> <p>What does they mean by <em>held rigid</em>?</p> </blockquote> <p>Later $C$ is a tree of $\mathbb P^1$'s. Let $G$ be the dual graph of $C$. Now they consider the space $Def_G(\mu) \subset Def(\mu)$ of first order deformations of the pointed stable map $\mu$ preserving the dual graph.</p> <blockquote> <p>What does they mean by <em>preserving the dual graph</em>? </p> </blockquote> <p>If I understand it correctly, $Def(\mu)$ consists of all $(\mathcal C \to Spec\ \mathbb C[\varepsilon], \bar p_i, \bar \mu: \mathcal C \to X)$ such that $\mathcal C \otimes_{\mathbb C[\varepsilon]} \mathbb C \cong C, \bar p_i \otimes \mathbb C = p_i$ and $C \to \mathcal C \xrightarrow{\bar \mu} X$ coincides with $\mu$. So, a geometric fiber of $\mathcal C$ should have the same dual graph as $C$.</p> <p>Thanks a lot.</p> http://mathoverflow.net/questions/99298/deformations-of-pointed-stable-maps-with-curve-held-rigid-or-preserving-the-du/99365#99365 Answer by Dan Petersen for Deformations of pointed stable maps with "curve held rigid" or "preserving the dual graph" Dan Petersen 2012-06-12T14:05:16Z 2012-06-12T14:05:16Z <p>Dear finite,</p> <p>First, a confusing typo in your short exact sequence. It should be $$0 \to H^0(C,T_C) \to \mathrm{Def}_R(\mu) \to \mathrm{Def}(\mu) \to 0.$$ What they mean by "held rigid" here is that they consider maps from $C = \mathbf P^1$ with a fixed choice of coordinates, not up to automorphism of $\mathbf P^1$. If we ignore marked points then the more "global" way of saying this is that $\mathrm{Def}(\mu)$ is the tangent space of <code>$\overline{M}_{0,0}(X,\beta)$</code> at the point $[\mu]$, whereas $\mathrm{Def}_R(\mu)$ is the tangent space of the Hom scheme $\mathrm{Hom}(C,X)$, which just parametrizes morphisms $C \to X$, at $[\mu]$. Naively, one expects then that <code>$\overline{M}_{0,0}(X,\beta)$</code> is locally the quotient of $\mathrm{Hom}(C,X)$ by $\mathrm{Aut}(C) \cong \mathrm{PGL}(2)$. This is essentially what the short exact sequence expresses, note e.g. that $\dim H^0(C,T_C) = \dim \mathrm{PGL}(2) = 3$.</p> <p>In the definition of $\mathrm{Def}_G(\mu)$ they just mean that they consider first order deformations whose dual graph is $G$. The corresponding "global" way of saying this is that they consider the locally closed substack of <code>$\overline{M}_{0,n}(X,\beta)$</code> defined by demanding that the dual graph is $G$, and consider the tangent space of this substack.</p>