It is well known that a smooth cubic surface $X\subset \mathbb{P}^3$ has exactly 27 lines in it. Furthermore, it is easy to check that Picard group $$Pic(X)\cong \mathbb{Z}^7$$ Here the generators are lines which are $\mathbb{P}^1$ topologically. Furthermore, it is easy to check that $$\chi\_{top}(X)=2+H\_2(X)=9$$Topologically speaking, notice that as smooth manifold $X$ has no 1-skeleton. This makes the 2-dimensional cells glue to points along their bounday, getting spheres $\mathbb{P}^1$ as the result of this gluing process. 


*My first question is **why the other 20 lines do not contribute to the Euler characteristic of $X$.***


Going further, if $X\subset \mathbb{P}^3$ has degree 4, it is known that $X$ sometimes has lines, sometimes it does not. However, $$\chi\_{top}(X)=2+H\_2(X)=24$$ meaning that, despite the fact that $X$ can perfectly have no lines, we still have homology $H\_2$ which are spheres topologically! meaning, there are in fact spheres (due to the argument above which says that $X$ has no 1-skeleton). Besides $\chi\_{top}$ is constant even though $X$ may have $64$, $32$ or even $0$ number of lines in it. There are spheres whose existence is not being noticed by $\chi\_{top}$ at all. Here let me be vague please. What is going on!?


Any type of editing to make this clearer will be welcome.
References highly appreciated.



Interaction of topology and the Picard group of Algebraic surfaces
Even if there is no lines or rational curves, there can still be spheres representing classes in $H^2$. There is no contradiction here since these spheres are maps from $S^2$ to the surface which are not necessarily holomorphic. Some of the spheres may be representable by holomorphic maps. Counting how many in each homology class (sometimes in some generalized sense) is a very interesting problem in enumerative geometry, which is connected to the theory of Mirror Symmetry. 


