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It has been proved (see [S. J. Kim, On the existence of Weierstrass gap sequences on trigonal curves. J. Pure Appl. Algebra  63  (1990), 171-180. MR1043748 (91b:14036) Zbl 0712.14019] and [M. Brundu, G. Sacchiero, On the varieties parametrizing trigonal curves with assigned Weierstrass points. Comm. Algebra  26  (1998), 3291-3312. MR1641619 (99g:14040) Zbl 0937.14016]. ) that a Weierstrass point on a trigonal curve, which is not a ramification point for the morphism induced by a trigonal series, has a gap sequence  G P   of the form  G P   = (1,2, …,  r ,  r  + 1 + α, …,  g  + α) for some integers  r  and α satisfying≤  r  ≤  g  − 1 and 1 ≤ α ≤ 2 r  + 1 −  g . Such points will be called one jump Weierstrass points of type ( r ,α). It was further proved [S. J. Kim, On the existence of Weierstrass gap sequences on trigonal curves. J. Pure Appl. Algebra  63  (1990), 171-180. MR1043748 (91b:14036) Zbl 0712.14019] that any one jump numerical sequence of type ( r , α), with  r  and α in the range above, is the Weierstrass gap sequence of a trigonal curve. Here we prove that the property of having an extremal, in some sense, one jump Weierstrass point characterizes trigonal curves. More precisely, we show that if α belongs to the range α 4  &lt; α ≤ 2 r  + 1 −  g  for a suitable α 4 , any curve with a Weierstrass point of type ( r ,α) is a triple covering of a smooth curve of genus  p  with, and that there exist examples of such coverings. Therefore when 2 r  + 1 −  g  − α &lt; 3, such a curve is indeed trigonal. As a consequence, any fourgonal curve of genus  g  ≥ 10 having a one jump Weierstrass point satisfies α ≤ α 4  with few exceptions. Finally, we exhibit examples of fourgonal curves with a Weierstrass point of type ( r , α) with≤ r ≤   and 1 ≤ α ≤ α 4 .

One jump Weierstrass gap sequence