## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 36, Number 2 (2013), 217-233.

### Weierstrass gap sequences at points of curves on some rational surfaces

Jiryo Komeda and Akira Ohbuchi

#### Abstract

Let $\tilde{C}$ be a non-singular plane curve of degree *d* ≥ 8 with
an involution σ over an algebraically closed field of characteristic 0 and
$\tilde{P}$ a point of $\tilde{C}$ fixed by σ. Let π :
$\tilde{C}$ → *C* = $\tilde{C}$/$/\langle\sigma\rangle $be
the double covering. We set *P* = π($\tilde{P}$). When the
intersection multiplicity at $\tilde{P}$ of the curve $\tilde{C}$ and the tangent line at $\tilde{P}$ is equal to *d* − 3 or
*d* − 4, we determine the Weierstrass gap sequence at *P* on *C*
using blowing-ups and blowing-downs of some rational surfaces.

#### Article information

**Source**

Tsukuba J. Math., Volume 36, Number 2 (2013), 217-233.

**Dates**

First available in Project Euclid: 21 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1358777000

**Digital Object Identifier**

doi:10.21099/tkbjm/1358777000

**Mathematical Reviews number (MathSciNet)**

MR3058240

**Zentralblatt MATH identifier**

1372.14028

**Subjects**

Primary: 14H55: Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx] 14H50: Plane and space curves 14H30: Coverings, fundamental group [See also 14E20, 14F35] 14J26: Rational and ruled surfaces

**Keywords**

Weierstrass gap sequence Weierstrass semigroup Smooth plane curve Double covering of a curve Blowing-up of a rational surface

#### Citation

Komeda, Jiryo; Ohbuchi, Akira. Weierstrass gap sequences at points of curves on some rational surfaces. Tsukuba J. Math. 36 (2013), no. 2, 217--233. doi:10.21099/tkbjm/1358777000. https://projecteuclid.org/euclid.tkbjm/1358777000