Let \(X \subset \mathbb P^N\) be an irreducible projective variety of dimension \(n\). J. M. Landsberg [J. Reine Angew. Math. 562, 1–3 (2003; Zbl 1041.51011)], proved that, if the number of lines contained in \(X\) and passing through a general \(x \in X\) is finite, then it is bounded by \(n!\).
The paper under review aims to obtain a similar bound for curves of degree \(d\); more generally, denoted by \(\text{Curves}_d(X,x)\) the space of curves of degree \(d\) lying on \(X\) and passing through a general \(x\) the author poses the following question: is it possible to give a bound on the number of components of \(\text{Curves}_d(X,x)\)? The answer to this question is affirmative, and in theorem 2 an explicit formula giving an upper bound in terms of \(n\) and \(d\) is given.
Another natural problem that arises in this context is the following: is it possible to give a bound also for non general points? The author shows by examples that this is not possible, but proves that away from a subvariety of codimension at least two such a bound, depending only on \(n\) and \(d\), exists.
The proof is based on effective bounds on the number of components of Chow varieties, on the study of the foliation on \(X\) generated by curves of degree \(d\), and on an argument in [L. Ein, O. Küchle and R. Lazarsfeld, J. Differ. Geom. 42, No. 2, 193–219 (1995; Zbl 0866.14004)].