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Gap Sequences of 1-Weierstrass Points on Non-Hyperelliptic Curves of Genus 10Q

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Gap Sequences of 1-Weierstrass Points on Non-Hyperelliptic Curves of Genus 10Q Journal of the Egyptian Mathematical Society (2014) 22, 317–321 Egyptian Mathematical Society Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems Gap sequences of 1-Weierstrass points on non-hyperelliptic curves of genus 10q Mohammed A. Saleem a,*, Eslam E. Badr b a Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt b Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt Received 20 July 2013; revised 18 November 2013; accepted 11 December 2013 Available online 25 January 2014 KEYWORDS Abstract In this paper, we compute the 1-gap sequences of 1-Weierstrass points of non-hyperelliptic 1-Weierstrass points; smooth projective curves of genus 10. Furthermore, the geometry of such points is classified as flexes, q-Gap sequence; sextactic and tentactic points. Also, upper bounds for their numbers are estimated. Flexes; Sextactic points; 2010 MATHEMATICS SUBJECT CLASSIFICATION: 14H55; 14Q05; 14Q99 Tentactic points; Canonical linear system; ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. Kuribayashi sextic curve Open access under CC BY-NC-ND license. 0. Introduction [3–8], the theory has been reformulated for Gorenstein curves, where the invertible dualizing sheaf substitutes the canonical Weierstrass points on curves have been extensively studied in sheaf. In this context, the singular points of a Gorenstein curve connection with many problems. For example, the moduli are always Weierstrass points. In [9], Notari developed a technique to compute the space Mg has been stratified with subvarieties whose points are isomorphism classes of curves with particular Weierstrass Weierstrass gap sequence at a given point, no matter it is points. For more deatails, we refer for example to [1,2]. simple or singular, on a plane curve, with respect to any linear 0 At first, the theory of the Weierstrass points was developed system V # H ðC; OCðnÞÞ. This technique can be useful to con- only for smooth curves and for their canonical divisors. In the struct examples of curves with Weierstrass points of a given last decades, starting from some papers by Lax and Widland weight or to look for conditions for a sequence to be a Weierst- rass gap sequence. He used this technique to compute the Wei- * Corresponding author. Tel.: +20 1064852498. erstrass gap sequence at a point of particular curves and of E-mail addresses: [email protected] (M.A. Saleem), families of quintic curves. [email protected] (E.E. Badr). The aim of this paper is to compute the 1-gap sequence of q arXiv: 1307.0078[math.AG]. the 1-Weierstrass points on smooth non-hyperelliptic algebraic Peer review under responsibility of Egyptian Mathematical Society. curves of degree 6, which are genus 10 curves, to investigate the geometry of such kind of points and to estimate an upper bound for the numbers of flexes, sextactic and tentactic points Production and hosting by Elsevier on such kind of algebraic curves. 1110-256X ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. Open access under CC BY-NC-ND license. http://dx.doi.org/10.1016/j.joems.2013.12.005 318 M.A. Saleem, E.E. Badr 1. Preliminaries (1) n1 ¼ 1. (2) nr 6 2r À 2 for every r P 2. Throughout this section, we use the following notations: ð1Þ 6 ðgÀ1ÞðgÀ2Þ (3) x ðpÞ 2 . (4) There are at least 2g þ 6 1-Weierstrass points on X. IðC ; C ; pÞ the intersection number of the curves C and C at the 1 2 1 2 For more details on q-Weierstrass points on Riemann point p [10], ðqÞ surfaces, we refer for example to [13,16]. Gp ðQÞ the q-gap sequence of the point p with respect to the linear system Q [11,12], xðqÞðpÞ the q-weight of the point p [13], 2. Main results NðqÞðCÞ the number of q-Weierstrass points on C [14], Qðð‘ Á pÞÞ the set of divisors in the linear system Q with Let X be a smooth projective plane curve of degree 6 and multiplicity at least ‘ at the point p [13]. Q :¼jKj the canonical linear system of X. Proposition 2.1. The linear system Q is g9 . r 18 Recall that a linear system Q is called a gd if dim Q ¼ r and deg Q d. We have the following result. ¼ Proof. The result follows directly from Lemma 1.1 [13]. Let Q be a nonempty gr linear system on an d dim Q :¼ dim jKj¼g À 1 ¼ 9; and algebraic curve X, and fix a point p 2 X. Then: deg ðQÞ :¼ deg ðKÞ¼2ðg À 1Þ¼18: à The set of gap numbers GpðQÞ is a finite set and Corollary 2.2. Let p 2 X, then ] Gð1ÞðQÞ ¼ 10 and jGpðQÞj ¼ 1 þ r. p Gð1ÞðQÞf1; 2; 3; ...; 19g. GpðQÞf1; 2; ...; 1 þ dg. p Lemma 2.3. The set of cubic divisors on X form a linear system 9 Let X be a smooth projective plane curve of genus g P 2 which is g18. and let D be a divisor on C with dimjDj¼r P 0. We denote by LðDÞ the C-vector space of meromorphic functions f such 2.1. Flexes that divðfÞþD P 0 and by lðDÞ the dimension of LðDÞ over C. Then, the notion of D-Weierstrass points [13] can be defined Definition 2.4. [11,17]. A point p on a smooth plane curve C is in the following way: said to be a flex point if the tangent line Lp meets C at p with contact order IpðC; Lp; pÞ at least three. We say that p is i-flex, Definition 1.2. Let p 2 C.Ifn is a positive integer such that if IpðC; LpÞ2 ¼ i. The positive integer i is called the flex order of p. lðD ðn À 1ÞpÞ > lðD À n Á pÞ; Our main results for this part are the following. we call the integer n a D-gap number at p. Theorem 2.5. Let p be a flex point on a smooth projective non- Lemma 1.3. Let p 2 C, then there are exactly r þ 1 D-gap hyperelliptic plane curve X of degree 6. Let Lp be the tangent lint numbers fn1; n2; ...; nrþ1g such that n1 < n2 < ÁÁÁ< nrþ1. The ð1Þ to X at p such that IðX; EpÞ¼lf. Then Gp ðQÞ sequence fn1; n2; ...; nrþ1g is called the D-gap sequence at p. ¼f1; 2; 3; 1 þ lf; 2 þ lf; 3 þ lf; 2lf þ 1; 2lf þ 2; 3lf þ 1; 3l P rþ1 f þ 2g. Moreover, the geometry of such points is given by Definition 1.4. The integer xDðpÞ :¼ i¼1 ðni À iÞ is called Table 1: D-weight at p.IfxDðpÞ > 0, we call the point p a D-Weierstrass point on C. In particular, for the canonical divisor K, the qK- Proof. The dimension of Qð1 Á pÞ and Qð2 Á pÞ does not Weierstrass points ðq P 1Þ are called q-Weierstrass points and p ðqÞ depend on whether is 1-Weierstrass point or not, i.e., the qK-weight is called q-weight, denoted by x ðpÞ. ð1Þ 1; 2 2 Gp ðQÞ. The spaces Qð3 Á pÞ¼...¼ Qðlf Á pÞ consist of divisors of cubic curves of the form LpR, where R is an arbi- Lemma 1.5. [13,15]. Let X be a smooth projective plane curve trary conic. Hence, dim Qð‘ Á pÞ¼6 for ‘ ¼ 3; ...; l . That is, ðqÞ f of genus g. The number of q-Weierstrass points N ðCÞ, counted 3 2 Gð1ÞðQÞ. with their q-weights, is given by p The space Q 1 p consists of divisors of cubic ðð þ lfÞ Þ gðg2 À 1Þ; if q ¼ 1 curves of the form LpR, where R is a conic passing through p. NðqÞðCÞ¼ ð2q À 1Þ2ðg À 1Þ2g; if q P 2: In particular, for smooth projective plane sextic (i.e. g ¼ 10), Table 1 Geometry of flexes. the number of 1-Weierstrass points is 990 counted with their ð1Þ ð1Þ xp ðQÞ Gp ðQÞ Geometry weights. 2 {1, 2, 3, ..., 8, 10, 11} 1-Flex 15 {1, 2, 3, 5, 6, 7, 9, 10, 13, 14} 2-Flex Theorem 1.6 [13]. Let X be a non-hyperelliptic curve of genus 28 {1, 2, 3, 6, 7, 8, 11, 12, 16, 17} 3-Flex ð1Þ P 3. Write Gp ðQÞ¼fn1 < n2 < ÁÁÁ< ngg, then Gap sequences of 1-Weierstrass points on non-hyperelliptic curves of genus 10 319 ð1Þ where i ¼ 1; 2; 3 and ½m is the greatest integer less than or equal Hence, dim Qðð1 þ lfÞpÞ¼5. That is, 1 þ lf 2 Gp ðQÞ. to m. The space Qðð2 þ lfÞpÞ consists of divisors of cubic curves of the form LpR, where R is a conic passing through p with contact order at least 2. Hence, dim Qðð2 þ lfÞpÞ¼4. That 2.2. Sextactic points ð1Þ is, 2 þ lf 2 Gp ðQÞ. The spaces Qðð3 þ lfÞpÞ¼ÁÁÁ¼Q ð2lf Á pÞ consist of divisors of cubic curves of the form In analogy with tangent lines and flexes of projective plane 2 curves, one can consider osculating conics and sextactic points LpH, where H is an arbitrary hyperplane. Hence, in the following way: dim Qð‘ Á pÞ¼3 for ‘ ¼ 3 þ lf; ...; 2lf. That is, 3 þ lf 2 Gð1ÞðQÞ. The space Qðð2l þ 1ÞpÞ consists of divisors of p f Lemma 2.9 [12].
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