I' m trying to prove that a curve in $\mathbb{P^{2}(\mathbb{C})}$ of degree $d$ has an infinite number of inflection points or it has at most $3d(d-2)$ inflection points.
Let be $C$ the curve and let be $H$ the Hessian curve of $C$. Suppose that the number of inflection points is finite but, by contradiction, greater than $3d(d-2)$. Then by Bezout's Theorem $C$ and $H$ have a common component. How can I proceed?
The set of nonsingular points in any variety is open and dense, because it is the nonvanishing set of the Jacobian determinant. Thus, any positive-dimensional variety over $\mathbb C$ contains infinitely many of them. As you pointed out, if $C$ and $H$ have a common component, then this is a positive dimensional variety all of whose nonsingular points are inflection. This is the contradiction you seek because there are infinitely many.