Curves of genus 3

1 Questions about curves (i) What is meant by the ‘number of points ’ on a curve? (ii) What is the number of points on a curve that can occur, given some parameters such as q, the size of the field, g, the genus of the curve, n, the degree of a plane curve? (iii) What is the maximum number of points? (iv) Find curves with certain parameters. (v) Classify the curves with a set of these parameters. 2 Starting point M.J. de Resmini Sulle quartiche piane sopra un campo di caratteristica due Ricerche di Matematica 19 (1970), 133–160 How many rational points are there on a non-singular plane quartic curve over Fq? 3 Cubic surface A non-singular surface F3 over a field K has at most 27 lines and over K exactly 27 lines. Over Fq, there exists an F3 with 27 lines if q 6 = 2, 3, 5. In PG(2, q), there exists a 6-arc not on a conic if q 6 = 2, 3, 5