Let $M$ denote the Kontsevich moduli space of stable maps $\overline M_{0,n}(X,\beta)$, where $\beta\in A_1(X)$ and $X$ is a convex variety. I am trying to understand why $$\dim\, M=\dim\,X+n-3+\int_\beta c_1(T_X).$$
First of all, $\int_\beta c_1(T_X)$ and $-K_X\cdot \beta$ are synonyms, where $K_X$ is the canonical class. Secondly, feel free to replace Chow classes in $A_i$ by homology classes in $H_{2i}$.
To give a point in $M$ is to give an isomorphism class of maps $f:C\to X$ where $C$ has genus $0$ and $n$ marks, and $f_\ast [C]=\beta$ at the level of Chow classes. Now, the curve $C$ plus its $n$ marks counts for $\dim \overline{M}_{0,n}=n-3$. I would like to understand why the map counts for the remaining $$\dim\,X-K_X\cdot\beta.$$
The truth is that I cannot realize what $\int_\beta c_1(T_X)=-K_X\cdot\beta$ is counting. So I would be quite satisfied if, beyond the above dimension count, one could also explain to me in general:
If $\gamma\in A_i(X)$ and $E$ is a vector bundle on $X$, what is $\int_\gamma c_i(E)$ counting?
Thank you!