Counting lines and conics on a surface (Q1035850)

The problem of studying the number of lines on surfaces in \(\mathbb{P}^3_{\mathbb{C}}\) is classical and it was started by \textit{B. Segre} [Q. J. Math., Oxf. Ser. 14, 86--96 (1943; Zbl 0063.06860); Proc. Lond. Math. Soc. (2) 49, 353--395 (1947; Zbl 0034.08603)]. Then many authors improved the bounds for the maximal number of lines on a surface in \(\mathbb{P}^3_{\mathbb{C}}\). The author study the problem from a more general point of view and he gives bounds for the number of rational curves and elliptic curves on complex pojective algebraic surfaces and he produce example of surfaces satisfying the upper bounds.