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Suppose that $K=D(A;B;D_1;D_2)$ be a boundry divisor of $\overline{M}_{0,n}(X,\beta)$ and let $\overline{M}_A=$ $\overline{M}_{0,A \cup \bullet}(X,\beta)$ where $\bullet$ is additional marking.let $e_A$ be evaluation map on $\bullet$.(Here $A$ is subset of $[n]$)

Now let $\overline{M}_A(X,\bar{t_A})$ be $\bar{t_A}$-rigid moduli space.In FP-notes by Fulton and Pandharipande at page 31 they define differential of $e_A$ at $[f]$ obtain as below:

They write we have a natural linear maps: $Def(f)\to H^0(f^*{T_X/T_C(-\bullet)}) \to T_X(f(\bullet))$

In the paper they imply that second map is obtained by natural fiber evaluation $H^0(f^*{T_X}) \to T_X(f(\bullet))$

I want to know why we have these two linear maps especially what is fiber evaluation map?

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  • $\begingroup$ Please add some details (notation, references) - as-is, your question is incomprehensible. $\endgroup$
    – Piotr Achinger
    Commented Oct 6, 2018 at 14:46
  • $\begingroup$ I wrote.these things are in FP-notes.You mean more refrences? $\endgroup$
    – Tom
    Commented Oct 6, 2018 at 14:49
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    $\begingroup$ I was able to guess what you meant by FP-notes, but it would be easier for everyone if you wrote out “the notes of Fulton and Pandharipande” and left a link: arxiv.org/abs/alg-geom/9608011 $\endgroup$
    – Samir Canning
    Commented Oct 6, 2018 at 14:59
  • $\begingroup$ As an aid to a possible answerer you could add in the body of the question the context. There is A LOT said in the paper about what you are asking. Could you tell us what you do understand from what is said there, and what specifically you do not understand? For example, right after the display you reproduce, it is said: "The first map in (36) is the natural surjection of Def(µ) onto the deformation space of the moduli problem obtained by forgetting all the markings except •. The natural fiber evaluation $H^0(\mu^*TX)\to TX(\mu(p\bullet))$.." etc. What exactly is clear and what unclear here? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 6, 2018 at 19:37
  • $\begingroup$ I cant understand comments that you implied and is written in the paper.I dont understand what here mean by fibre evaluation and why we have that surjection for first map. $\endgroup$
    – Tom
    Commented Oct 6, 2018 at 19:46

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