One jump Weierstrass gap sequence

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 ≤ 2r + 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 α₄ < α ≤ 2r + 1 − g for a suitable α₄, 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 2r + 1 − g − α < 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 ≤ α₄ with few exceptions. Finally, we exhibit examples of fourgonal curves with a Weierstrass point of type (r, α) with ≤ r ≤ and 1 ≤ α ≤ α₄.

Tools

Send paper

Add to favourites

Publisher and Institutes

You are here: Home :: Area NEM :: Mathematics :: Geometry

V Beorchia, G Sacchiero

One jump Weierstrass gap sequence

Tools

Advances in Geometry, Walter de Gruyter