Suppose that $C\cong P^1$ and $Def(f)$ denote the first order deformation of pointed stable map $(C,{p_i},f:C\longrightarrow X)$. I read that we have short exact sequence:

$0\longrightarrow H^0(C,T_C)\longrightarrow Def_R(f)\longrightarrow Def(f) \longrightarrow 0$

Where $Def_R(f)$ is the first order deformation of $(C,{p_i},f:C\longrightarrow X)$ with $C$ held rigid.

1)what does it mean(($C$ held rigid)? Does it mean we consider $C$ fixed?

2)Why we have this short exact sequence?

(If f is closed immersion and ignore marking and my comment on 1 is correct we have $0\longrightarrow T_C \longrightarrow f^*T_X \longrightarrow N_f\longrightarrow 0$ so we will get the short exact sequence because $P^1$ is rigid.But in general i cant reach to this short exact sequence )

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    $\begingroup$ It would help if you explain the reference you are reading and what is your background. For experts in deformation theory, a citation to a page number in Illusie's book might be sufficient, but for others, it might be helpful to read a textbook like Cox-Katz or a survey like Fulton-Pandharipande. $\endgroup$ – Jason Starr Sep 22 '18 at 21:04
  • $\begingroup$ I am reading Fulton-panharipande note and they write this exact sequence. $\endgroup$ – Tom Sep 22 '18 at 21:10
  • $\begingroup$ Forgetting the marked points, you can think about deformations of the "map" holding the domain and target fixed as deformations of the graph of the map inside the product of the domain and the target. The graph of the map is a closed curve inside the product of the domain and the target. By deformation theory of the Hilbert scheme / Chow scheme / deformations of an immersed smooth subvariety (or whatever your background is), the first-order deformation space and obstruction space are $H^0$ and $H^1$ of the pullback of the tangent bundle. $\endgroup$ – Jason Starr Sep 22 '18 at 21:28
  • $\begingroup$ But what does (C held rigid) means? $\endgroup$ – Tom Sep 22 '18 at 21:44
  • $\begingroup$ The deformations of the graph of a morphism as a closed subscheme of the product and the target are "deformation of the morphism with the domain the target held fixed". $\endgroup$ – Jason Starr Sep 22 '18 at 22:53

The comment thread is getting too long. There are several good introductions to infinitesimal deformation theory and obstruction theory for various functors / stacks in groupoids. One source that I like is the first chapter of Kollár's Rational curves on algebraic varieties. In that chapter he discusses the infinitesimal deformation theory of the Hilbert scheme and the infinitesimal deformation theory of the Hom scheme. If you understand these two theories for isomorphisms from $\mathbb{P}^1$ to smooth conics in $X=\mathbb{P}^2$, then you will see that the exact sequence above "must be" correct.

The original proofs of results in infinitesimal deformation theory go back, at least, to Kodaira and Spencer, and then to Grothendieck's 2-term truncation of the cotangent complex, to work of Schlessinger, to work of Rim, to Lichtenbaum-Schlessinger, to André and Quillen, etc. The reference I like is Illusie's "Complexe cotangents et déformations."


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