Tangent space of the stack $\overline{\mathcal{M}}_{g,n}(X,\beta)$. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:38:43Z http://mathoverflow.net/feeds/question/114963 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114963/tangent-space-of-the-stack-overline-mathcalm-g-nx-beta Tangent space of the stack $\overline{\mathcal{M}}_{g,n}(X,\beta)$. Zheng 2012-11-30T08:42:45Z 2012-12-02T23:29:43Z <p>Let $\overline{\mathcal{M}}_{g,n}(X,\beta)$ be the moduli stack of stable maps $f$ from genus $g$, $n$-marked curve $C$ to a variety $X$ i nth curve class $\beta \in H^2(X,\mathbb{Z})$. I would like to know why the hyperext group $$Ext^1_C(f^*\Omega_X\rightarrow \Omega_C(\sum_{i=1}^n),\mathcal{O}_C)$$ represents tangent space of the moduli and the obstruction lies in<br> $$Ext^2_C(f^*\Omega_X\rightarrow \Omega_C(\sum_{i=1}^n),\mathcal{O}_C)?$$ I am aware that the deformation-obstrcution of a map $f:C\rightarrow X$ with a fixed curve $C$ is governed by $H^i(C,f^*T_X)$ for $i=1,2$ and the automorphism-deformation of a $n$-marked points $(C;p_1,\dots,p_n)$ is governed by $Ext^i_C(\Omega_C(\sum_{i=1}^n),\mathcal{O}_C)$ for $i=0,1$. However, I don't know how to combine them into one package in the hyperext groups above. </p> <p>I would appreciate it if someone could kindly explain how to obtain and understand the hyperext groups above. </p> http://mathoverflow.net/questions/114963/tangent-space-of-the-stack-overline-mathcalm-g-nx-beta/115203#115203 Answer by Barbara for Tangent space of the stack $\overline{\mathcal{M}}_{g,n}(X,\beta)$. Barbara 2012-12-02T21:20:15Z 2012-12-02T21:20:15Z <p>The classical reference is Illusie's PhD thesis "complexe cotangent et deformations".There the result without the marked points is proven in a very, very general context using the cotangent complex; then one needs to know that for a smoooth variety, or for a nodal curve (or, again, in many more cases) the cotangent complex is isomorphic in the derived category to the cotangent sheaf. </p> <p>Fixing the case of the marked points can easily be done by hand (they're smooth points after all): I can expand on this step if it helps. Or, one can go modern and use logarithmic geometry. </p> <p>If you want to get a feeling for why such a result is true, a very short reference is a paper by Ziv Ran on deformations of morphisms: Algebraic Curves and Projective Geometry Lecture Notes in Mathematics Volume 1389, 1989, pp 246-253 Deformations of maps</p> <p>Another approach would be to sit down and prove it yourself using as only references a few classical facts about deformation theory (say, what you learn from the first ten pages of Artin's Tata lectures). Whether this is a useful exercise or a waste of time depends on your personality. Again, details upon request.</p> http://mathoverflow.net/questions/114963/tangent-space-of-the-stack-overline-mathcalm-g-nx-beta/115215#115215 Answer by Jonathan Wise for Tangent space of the stack $\overline{\mathcal{M}}_{g,n}(X,\beta)$. Jonathan Wise 2012-12-02T23:29:43Z 2012-12-02T23:29:43Z <p>Let $L$ be the complex $[ f^\ast \Omega_X \rightarrow \Omega_C(D) ]$, concentrated in degrees $[-1,0]$ on $C$.</p> <p>One way to understand the obstructions is to understand the deformations for affine curves first. Observe first that there is a canonical equivalence of categories between the category of deformations of a pointed map $f : C \rightarrow X$, with $C$ not-necessarily proper, and the category of extensions</p> <p>$G(C) = \mathrm{Ext}(L, \mathcal{O}_C)$.</p> <p>The meaning of the category on the right is the category of extensions $E$ of $\Omega_C(D)$ by $\mathcal{O}_C$ together with a trivialization of the induced extension of $f^\ast \Omega_X$ by $\mathcal{O}_C$. One checks explicitly that the category of extensions of $\Omega_C(D)$ by $\mathcal{O}_C$ is canonically equivalent to the category of deformations of the pointed curve $C$ and that splitting the induced extension of $f^\ast \Omega_X$ is the same as extending the map $f : C \rightarrow X$ to a map from the deformed curve to $X$.</p> <p>Note that $G$ is a commutative group stack (what is sometimes called a "Picard stack"). It is easy to check that $G$ is non-empty when $C$ is an affine curve (use the fact that deformations of nodal curves are unobstructed and the formal criterion of smoothness for $X$), which means that any obstructions to deforming a proper curve come from the failure of local deformations to glue. There is a standard way to compute these local obstructions---e.g., by Cech cohomology, noting that mutatis mutandis, Cech cohomology still works for group stacks---and this yield a class in $H^1(C, G)$. Cohomology of a commutative group stack is the same as cohomology of the associated complex, so we get an obstruction in</p> <p>$H^1(C, L^\vee[1]) = H^2(C, L^\vee) = \mathrm{Ext}^2(L, \mathcal{O}_C)$.</p>