# Smooth Curve that intersects any degree d hypersurface at finitely many points

Fix an integer $d>0$. Can we find a smooth curve in complex projective space ${\mathbb{CP}}^n$ that does not lie in the zero locus of any polynomial of degree $d$?

In terms of polynomials, this means the following. Can we find $n-1$ polynomials $p_1,\ldots, p_{n-1}$ such that for every polynomial $q$ of degree $d$, the system of equations $p_1=\cdots = p_{n-1}=q=0$ has only finitely many solutions? (I assume these polynomials $p_i$ are generic enough so that their intersection is smooth.)

-
Just take a general complete intersection of dimension $1$ of hypersurfaces of degree strictly greater than $d$. It is known (and easily proven by induction on the number of hypersurfaces) that the complete linear system of hypersurfaces of degree $d$ injects into the linear system on the curve. – Torsten Ekedahl Nov 6 '10 at 9:01
Another way to see the same thing: Veronese embed P^n into P^m as a spanning variety X, by a basis of monomials of degree d. Then all hypersurfaces of degree d in P^n are pull backs of hyperplanes of P^m. So we want a curve in X that does not lie on any hyperplane of P^m. So choose a finite set of points on X that span P^m. Then choose a curve on X that passes through all these points. [This can be found by blowing up all these points, then re embedding X' in projective space, and choosing a general complete intersection of hyperplanes that cut on X' a smooth irreducible curve via Bertini. – roy smith Dec 13 '10 at 3:13