Weierstrass Pure Gaps on Curves With Three Distinguished Points

Let
$ \mathbb K$
be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree
$n+1>3$
over
$ \mathbb K$
, containing three points,
$P_{1},P_{2}$
, and
$P_{3}$
, such that
$nP_{1}+P_{2}$
,
$nP_{2}+P_{3}$
, and
$nP_{3}+P_{1}$
are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on
$\{P_{1},P_{2},P_{3}\}$
, as well as an explicit description of all pure gaps at each nonempty subset of the distinguished points
$P_{1},P_{2}$
, and
$P_{3}$
. When
$ \mathbb K=\overline { \mathbb F}_{q}$
, this class of curves, which includes the Hermitian curve, is used to construct algebraic geometry codes having minimum distance better than the Goppa bound.