Consider $X\subset\mathbb{P}^n$ smooth algebraic variety of degree $d$. I want to prove that the set of lines $F=\{l : l\subset X\}$ is a projective variety.
As far as I understand, I need to show, that there exists a finite set of polynomials, which vanishes on my lines. Also I know that there is a finite number of polynomials which describes $X$.
Probably we can substitute parametric equations for line to the system of polynomials of $X$ and this system will discribe all lines?
I know that it is a Fano variety of lines (by the definition of Fano variety from Harris). So generally, why Fano variety is a variety?
Thank you.
The variety of lines in $X \subset \mathbb{P}(V)$ is a subvariety of the Grassmannian $Gr(2,V)$. If $f_1,\dots,f_m$ are homogeneous equations (of degrees $d_1,\dots,d_m$) defining $X$, they provide sections of the vector budndles $S^{d_1}\mathcal{U}^\vee,\dots,S^{d_m}\mathcal{U}^\vee$, where $\mathcal{U}$ is the tautological bundle. Their common zero locus is the Fano variety of lines on $X$. As the zero locus of a global section of a vector bundle, it has a natural scheme structure.