# One jump Weierstrass gap sequence

One jump Weierstrass gap sequence Abstract 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 < 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 < 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 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

# One jump Weierstrass gap sequence

, Volume 6 (1) – Jan 26, 2006
13 pages

/lp/de-gruyter/one-jump-weierstrass-gap-sequence-kyUmgC1qpK
Publisher
de Gruyter
ISSN
1615-715X
eISSN
1615-7168
DOI
Publisher site
See Article on Publisher Site

### Abstract

Abstract 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 < 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 < 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 .