# Prove that the set of lines on smooth variety is a variety

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.

• Do you mean $l \subset X$ ? – Nicolas Hemelsoet Feb 18 at 10:23
• Exactly. Thanks – Kirill Losev Feb 18 at 10:26

## 1 Answer

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.

• Thank you, but this is far too difficult. Is there any simpler way? – Kirill Losev Feb 18 at 11:50
• Why is it difficult? – Sasha Feb 18 at 11:54
• I haven't studied vector bundles and schemes yet. – Kirill Losev Feb 18 at 12:00
• So, this is a good reason to study them! – Sasha Feb 18 at 12:10
• If I recall correctly, Eisenbud and Harris write down equations for the Fano variety in terms of the equations of the underlying variety. – aginensky Feb 18 at 16:09