A bound on the number of curves of a given degree through a general point of a projective variety

Let $X$ be an irreducible projective variety of dimension $n$ in a projective space and let $x$ be a point of $X$. Denote by ${\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e}\mathrm{s}}_d(X,x)$ the space of curves of degree $d$ lying on $X$ and passing through $x$. We will show that the number of components of ${\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e}\mathrm{s}}_d(X,x)$ for any smooth point $x$ outside a subvariety of codimension $\ge 2$ 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.