On the geometry of Hermitian one-point codes

Here we describe the algebraic geometry of one-point codes from the Hermitian curve. In particular, we employ zero-dimensional schemes in the plane to characterize the minimum-weight codewords of their dual codes, providing explicit formulas for their number. We discuss also some natural improvements of the duals of Hermitian one-point codes by means of geometric arguments. Finally, some cohomological tools are developed to characterize the small-weight codewords of such codes.