  Let $X$ be an irreducible projective variety of dimension $n$ in a projective
space and let $x$ be a point of $X$. Denote by ${\rm Curves}_d(X,x)$ the space
of curves of degree $d$ lying on $X$ and passing through $x$. We will show that
the number of components of ${\rm Curves}_d(X,x)$ for any smooth point $x$
outside a subvariety of codimension $\geq 2$ is bounded by a number depending
only on $n$ and $d$. An effective bound is given. A key ingredient of the proof
is an argument from Ein-K\"uchle-Lazarsfeld's work on Seshadri numbers.
