Quintic equation and number of lines on the quintic I heard a talk where the speaker said that the solution to the equation
$x_1^5 +x_2^5 +x_3^5 +x_4^5 +x_5^5 = 0$
is a six-dimensional (Calabi-Yau) manifold. Then he went on to define five curves of degree d:
$x_i = a_{i,d} z^d + a{_i,d-1} z^{d-1} + ... a_{i,0}$ where i goes from 1 to 5.
He now says that there are 2875 lines (curves of degree d=1) on the manifold given above.
I have a number of questions:
i) How does the given quintic equation yield a six-dimensional manifold if we only have five variables?
ii) Where does the number 2875 come from? I'm not interested in a rigorous  mathematical derivation as I'll probably not understand it anyway (not a mathematician, just an enthusiast) but I'm surprised about the fact that there actually is a finite number of curves on that manifold. I mean, if my coefficients $a_{i,*}$ can be any real number, shouldn't we  get a continuous family of curves?
 A: The equation is for points of complex projective space, so the variables $x_1, \dotsc, x_5$ are complex numbers, not all $0$, and $(x_1, \dotsc, x_5)$ and $(\lambda x_1, \dotsc, \lambda x_5)$ denote the same point for $\lambda \in \mathbb{C} - 0$. So even though we use $5$ variables, there is some redundancy and this space is only $4$ complex dimensional. Now imposing that equation cuts down the dimension by $1$ to give a complex $3$-manifold, which is $6$-dimensional over $\mathbb{R}$ because the complex numbers are $2$-dimensional over $\mathbb{R}$.
For your other question, it is very important that we only count lines and not all curves. For comparison, take the $2$-dimensional sphere $\{x^2 + y^2 + z^2 = 1\}$ inside $\mathbb{R}^3$. This doesn't contain any straight lines. On the other hand, if you consider the cubic surface $\{x^3 + y^3 + z^3 = 1\}$ then that contains every point of the line $\{x = -y, z = 1\}$. In fact there are only $3$ lines with $\mathbb{R}$-coefficients that are contained in this surface, while if you take complex valued $x, y, z$ there are $27$ complex "lines" (1-dimensional solution sets of linear equations). If you perturb these lines slightly you can get curves that are contained in the surface but they will not be given by linear equations.
