Suppose we have family $(\pi:U \longrightarrow X), \{p_i\},f)$ be in $\overline{M}_{g,n}(P^r,d).$ I want to know what does the following statement mean:
the cartier divisors $f^\ast{t_i}$ split into sections $\{q_i,_j\}$
We have $f:U\longrightarrow P^r$. I think it means that when we look at a fibre of $x$ which is a stable curve then zero scheme of$f^\ast{t_i}$ as a cartier divisor is $\{q_{i,1}(x)\} +\{q_{i,2}(x)\} +\cdots +\{q_{i,d}(x)\}$.
$t_0$,$t_1$,...,$t_r$ is basis for $H^0(P^r,O(1))$.
Am i right?