Dependent rational points on curves over finite fields - Lefschetz theorems and exponential sums

For an algebraic curve defined over Fq we study the probability that t randomly chosen Fq-rational points on the curve impose dependent conditions on the functions in a given τ-dimensional vector space of rational functions on the curve. This probability tends to be close to 1q. The results have applications in the assessment of the performance of decoding algorithms for algebraic geometry codes. The proofs involve a geometric construction. Lefschetz theorem for quasi-projective varieties and majorizations of exponential sums.