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.

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

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.

  • $\begingroup$ Thank you, but this is far too difficult. Is there any simpler way? $\endgroup$ – Kirill Losev Feb 18 at 11:50
  • $\begingroup$ Why is it difficult? $\endgroup$ – Sasha Feb 18 at 11:54
  • $\begingroup$ I haven't studied vector bundles and schemes yet. $\endgroup$ – Kirill Losev Feb 18 at 12:00
  • 3
    $\begingroup$ So, this is a good reason to study them! $\endgroup$ – Sasha Feb 18 at 12:10
  • 1
    $\begingroup$ If I recall correctly, Eisenbud and Harris write down equations for the Fano variety in terms of the equations of the underlying variety. $\endgroup$ – aginensky Feb 18 at 16:09

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