I can't understand the following assertion, from Shafarevic,Basic Algebraic Geometry,Vol 1,pag 80.

It is easy to construct a cubic surface $X\subset\mathbb{A}^3$ not containing lines. For example, if $X$ is given by

$$T_1T_2T_3=1$$ then $X$ does not have a single line contained in $\mathbb{A}^3$. Indeed, if we write the equation of an affine line in the form $T_i=a_it+b_i$ for $i=1,2,3$ and substitute in the equation above, we get a contradiction.

I did the substituion and I got the following relation


Well, where is the contradiction?


1 Answer 1


It is the whole line, so it has to be true for all $t$. So the left-hand side has to have the coefficients of $t^3$, $t^2$ and $t$ equal to 0, and the constant term equal to 1.

Since $a_1a_2a_3=0$, you need $a_1=0$ say. Can you take it from there?


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