An Exponential Sum and Higher-Codimensional Subvarieties of Projective Spaces over Finite Fields

A general method to express in terms of Gauss sums the number of rational points of subschemes of projective schemes over finite fields is applied to the image of the triple embedding ${\mathbb{P}}^1\hookrightarrow {\mathbb{P}}^3$. As a consequence, we obtain a non-trivial description of the value of a Kloosterman-sum-like exponential sum.