In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-5 curves that are neither hyperelliptic nor trigonal. Subsequently, we construct an algorithm for a complete enumeration of non-special genus-5 curves having more rational points than a specified threshold, where “non-special curve” means that the curve is non-hyperelliptic and non-trigonal with mild singularities of the associated sextic model that we propose. As a practical application, we implement this algorithm using the computer algebra system MAGMA, specifically for curves over the prime field of characteristic 3.

In algebraic geometry, the Frobenius map (or called Frobenius action, or Frobenius operator) F⁎ on cohomology groups plays an important role in the classification of algebraic varieties over a field of positive characteristic. In particular, representation matrices for F⁎ give rise to many important invariants such as p-rank and a-number. Several methods for computing representation matrices for F⁎ have been proposed for specific curves. In this paper, we present an algorithm to compute representation matrices for F⁎ of general projective schemes over a perfect field of positive characteristic. We also propose an efficient algorithm specific to complete intersections; it requires to compute only certain coefficients in a power of a multivariate polynomial. Our algorithms shall derive fruitful applications such as computing Hasse-Witt matrices, and enumerating superspecial curves. In particular, the second algorithm for complete intersections provides a useful tool to judge the superspeciality of an algebraic curve, which is a key ingredient to prove main results in Kudo and Harashita, 2017a, Kudo and Harashita, 2017b, Kudo and Harashita, 2020 on the enumeration of superspecial genus-4 curves.