This is a problem of Harris, Algebrai Geometry, A First Course, Ex. 1.3.
It says $\Gamma\subset P^n$ contains d points which are not contained in a line, then $\Gamma$ can be described as the zero locus of polynomials of degree less than or equal $d-1$.
I tried the following. However, I could not lower the degree of polynomials to d-1 or less.
Now the line in $P^n$ is a plane through origin of $C^{n+1}$. It suffices to solve the problem when the plane coincide on $x_0$-$x_1$ plane in $C^{n+1}$ as we can always rotate the plane. Now we see that $\Gamma\subset U_2=\{[s_1,s_2,1,...,s_n]=[\frac{p_0}{p_2},\frac{p_1}{p_2},1,\frac{p_3}{p_2},...,\frac{p_n}{p_2}]\in P^n|p_2\neq 0\}$ as $\Gamma$ does not intersect the plane. It is clear that $\Gamma$ is projective variety. So $\Gamma\cap U_2$ is affine variety. However, we have $d$ points on which polynomials vanishes. Since $C$ is algebraically closed, obviously, we have polynomials of degree $d$ as we have a product of the form $\prod_j(s_i-\frac{p_{ij}}{p_{2j}})$ for $j=1,...,d$ where i is the $i-th$ coordinate different from 2.
What is wrong with the reasoning above?