Consider the family of curves defined by $f(x,y)=g(x)+h(y)+a$, where $a$ is a free parameter. Now, it states that the family of curves intersect at $\infty$ and that $\infty$ is a base-point of these family of curves. What is the definition/meaning of base-point in this context?
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Given a linear system and any point $x ∈ X$, $X$ a $k$-scheme, on which all elements of the linear system $V$ vanish, we say that $x$ is a base-point of $V$. See here.
Suppose $f_a(x,y)=g(x)+h(y)+a=\sum_{i=0}^N b_ix^i+ \sum_{j=0}^n c_jx^j +a $ with, say, $N\geq n$ and $b_N\neq 0$ .
Homogeneizing we get $F_a(x,y,z)=\sum_{i=0}^N b_ix^iz^{N-i}+ \sum_{j=0}^n c_jx^jz^{N-j} +az^N$.
The points at infinity of the curve $C_a=V(F_a)$ obtained by setting $F_a(x,y,z)=0$ are the points $\infty_k=(\xi_k:\eta_k:0)$ where the $(\xi_k:\eta_k)$ are the zeros of the equation $b_Nx^N+c_Ny^N=0$ (with the obvious convention that $c_N=0$ if $n\lt N$).
Have you noticed that the $\infty_k=(\xi_k:\eta_k:0)$ do not depend on $a$ ?
This means that all the curves $C_a=V(F_a)$ go through these infinity points $\infty_k$ and this is the meaning of the $\infty_k$'s being base points of the family (=linear system= pencil) of curves $(C_a)_a$ .