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I am going through Andreas Gathmann's notes on algebraic geometry. On the first chapter, where he provides some motivation to the study of algebraic geometry, he gives as an exercise the following question:

Let $S ⊂ \mathbb{C}^3$ be a cubic surface, i. e. the zero locus of a polynomial of degree 3 in the three coordinates of $C^3$. Find an argument why you would expect there to be finitely many lines in $S$ (i. e. why you would expect the dimension of the space of lines in $S$ to be 0-dimensional). What would you expect if the equation of $S$ has degree less than or greater than 3?

I am just starting my algebraic geometry class, so I don't know a lot of algebraic geometry- just about the Zariski topology, Hilbert's zeros theorem and all the stuff you cover in the first couple of lecturs. So I am not looking for a mathematically rigorous solution, but rather some huristic argument to understand this fact.

I was thinking along this line - a line $l$ is an algebraic variety $V(I)$. As $l\subset S$, we know that if $S=Z(<f>)$, then $f\in I$. But here I got stuck as $I$ is generated by 2 linear polynomials and I don't see how can I say anything about these polynomials from the fact that they generate $f$ together. Maybe it is just not the direction.

I will be happy for any hints on how to approach this or any explanations of the fact.

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I think what he is getting at is to compare your answers to the following questions:

--- given a fixed line $l$ in $\mathbf C^3$, how many linear conditions are imposed on the coefficients of a cubic polynomial by requiring that the polynomial vanish identically on $l$?

--- what is the dimension of the space of lines in $\mathbf C^3$?

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  • $\begingroup$ Thank you! It helps a lot :) $\endgroup$ – Madarb Oct 28 '18 at 5:06

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