Journal of the Egyptian Mathematical Society (2014) 22, 317–321
Egyptian Mathematical Society Journal of the Egyptian Mathematical Society
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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 2 for every r P 2. Throughout this section, we use the following notations: ð1Þ 6 ðg 1Þðg 2Þ (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], Qð ð‘ 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 1 ¼ 9; and algebraic curve X, and fix a point p 2 X. Then: deg ðQÞ :¼ deg ðKÞ¼2ðg 1Þ¼18: 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 Gð1ÞðQÞ f1; 2; 3; ...; 19g. GpðQÞ f1; 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; LpÞ 2 ¼ i. The positive integer i is called the flex order of p. lðD ðn 1Þ pÞ > lðD n 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 < < 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 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 Qð 1 pÞ and Qð 2 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 Qð 3 pÞ¼...¼ Qð lf 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 ‘ 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 ð ð þ lfÞ Þ gðg2 1Þ; if q ¼ 1 curves of the form LpR, where R is a conic passing through p. NðqÞðCÞ¼ ð2q 1Þ2ðg 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 < < 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 Qð ð1 þ lfÞ pÞ¼5. That is, 1 þ lf 2 Gp ðQÞ. to m. The space Qð ð2 þ lfÞ pÞ 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 Qð ð2 þ lfÞ pÞ¼4. That 2.2. Sextactic points ð1Þ is, 2 þ lf 2 Gp ðQÞ. The spaces Qð ð3 þ lfÞ pÞ¼ ¼Q ð 2lf 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 ‘ pÞ¼3 for ‘ ¼ 3 þ lf; ...; 2lf. That is, 3 þ lf 2 Gð1ÞðQÞ. The space Qð ð2l þ 1Þ pÞ consists of divisors of p f Lemma 2.9 [12]. Let p be a non-flex point on a smooth 2 cubic curves of the form LpH, where H is a hyperplane through projective plane curve X of degree d P 3. Then there is an ð1Þ p. Hence, dim Qð ð2lf þ 1Þ pÞ¼2. That is, 2lf þ 1 2 Gp unique irreducible conic Dp with IpðX; Dp; pÞ P 5. This unique irreducible conic D is called the osculating conic of X at p. ðQÞ. The spaces Qð ð2lf þ 2Þ pÞ¼ ¼Q ð 3lf pÞ consist p 2 of divisors of cubic curves of the form LpH, where H is a Definition 2.10 [11]. A non-flex point p on a smooth projective hyperplane through p with contact order at least 2. Hence, ð1Þ plane curve X is said to be a sextactic point if the osculating dim Qð ð2lf þ 2Þ pÞ¼1. That is, 2lf þ 2 2 Gp ðQÞ. The conic Dp meets X at p with contact order at least six. A sextac- space Qð ð3l þ 1Þ pÞ consists of the divisor of the cubed f tic point p is said to be i-sextactic,ifIpðX; Dp; pÞ 5 ¼ i. The 3 tangent line Lp. Hence, dim Qð ð3lf þ 1Þ pÞ¼0. That is, positive integer i is called the sextactic order. ð1Þ 3lf þ 1 2 Gp ðQÞ. The spaces Qð ‘ pÞ¼/ for ‘ P 3lf þ 2. Now, the main results for this part are the following. ð1Þ That is, 3lf þ 2 2 Gp ðQÞ. Consequently, Theorem 2.11. Let p be a sextactic point on a smooth projective ð1Þ Gp ðQÞ¼f1; 2; 3; 1 þ lf; 2 þ lf; 3 þ lf; 2lf þ 1; 2lf þ 2; 3lf non-hyperelliptic curve X of degree 6. Let Dp be the osculating conic to X at p such that IðX; Dp; pÞ¼ls. Then, þ 1; 3lf þ 2g: ð1Þ Gp ðQÞ¼f1; 2; 3; ...; 7; 1 þ ls; 2 þ ls; 3 þ lsg. Finally, by the famous Bezout’s theorem, the tangent line 6 6 meets X at the flex point p with 3 lf 6. On the other hand, Proof. The idea of the proof is to investigate the existence 6 by Theorem 1.6, it follows that n10 :¼ 3lf þ 2 18, hence of a curve H through p with multiplicity ‘ so that its divisor – h lf 6. is in Qð ‘ pÞ Qð ð‘ þ 1Þ pÞ, consequently, the integer ð1Þ ‘ 2 Gp ðQÞ. Now, the dimension of Qð 1 pÞ and Qð 2 pÞ Corollary 2.6. If a smooth projective curve X of degree 6 has does not depend on whether p is a 1-Weierstrass point or 4-flex points, then X is hyperelliptic. ð1Þ not, i.e., 1; 2 2 Gp ðQÞ. Moreover, let Lp be the tangent line to X at p, then the divisor divðLpR0Þ2Qð 2 pÞ Qð 3 pÞ, Corollary 2.7. On a smooth non-hyperelliptics projective plane where R is a conic not through p. That is, 3 2 Gð1ÞðQÞ. Fur- curve X of degree 6, the 1-weight of a flex point is given by 0 p ð1Þ thermore, div LpR1 2 Qð 3 pÞ Qð 4 pÞ, where R1 is a xp ðQÞ¼13lf 37, where lf is the multiplicity of the tangent conic passing through p with multiplicity 1. That is, line Lp to X at p. ð1Þ 2 4 2 Gp ðQÞ. Also, div LpH0 2 Qð 4 pÞ Qð 5 pÞ, where Proof. Let p be a flex point on X, then, by Theorem 2.5, H is a hyperplane not through p. That is, 5 Gð1Þ Q . Simi- 0 2 p ð Þ ð1Þ larly, div L2H Q 5 p Q 6 p , where H is a hyper- Gp ðQÞ¼f1;2;3;1 þ lf;2 þ lf;3 þlf;2lf þ 1;2lf þ 2;3lf þ 1;3lf þ2g: p 1 2 ð Þ ð Þ 1 plane passing through p with multiplicity 1. That is, Consequently, ð1Þ 6 2 Gp ðQÞ. ð1Þ g xp ðQÞ : ¼ Rr¼0ðnr rÞ¼ð1 þ lf 4Þþð2 þ lf 5Þ Now, the spaces Qð 7 pÞ¼...¼ Qð ls pÞ consist of þð3 þ lf 6Þþð2lf þ 1 7Þþð2lf þ 2 8Þ divisors of cubic curves of the form DPH, where H is an arbitrary ð1Þ þð3lf þ 1 9Þþð3lf þ 2 10Þ¼13lf 37: line. Hence, 7 2 Gp ðQÞ. On the other hand, the space q Qð ð1 þ l Þ pÞ consists of divisors of cubic curves of the form Notation. Fð ÞðXÞ will denote the set of i-flex points which are s i D H, where H is a hyperplane through p. Consequently, q-Weierstrass points on X and NFðqÞ X will denote the cardi- P i ð Þ ð1Þ ðqÞ 1 þ ls 2 Gp ðQÞ. Also, the space Qð ð2 þ lsÞ pÞ contains only nality of Fi ðXÞ. ð1Þ the cubic divisor DPLp. Then, 2 þ ls 2 Gp ðQÞ. Finally, Corollary 2.8. For a smooth non-hyperelliptic projective plane Qð ‘:pÞ¼/, for ‘ P 3 þ l . That is, 3 þ l 2 Gð1ÞðQÞ. h ð1Þ s s p curve X of degree 6, the maximal cardinality of Fi ðXÞ is given by the inequality Corollary 2.12. Let p be a sextactic point on a smooth projective non-hyperelliptic curve X of degree 6. Then, the geometry of such 990 NFð1ÞðXÞ 6 ; points is given by Table 2: i 13ði þ 2Þ 37 320 M.A. Saleem, E.E. Badr
Table 2 Geometry of sextactic points. ð1Þ ð1Þ ð1Þ ð1Þ xp ðQÞ Gp ðQÞ Geometry xp ðQÞ Gp ðQÞ Geometry 1 {1,2,3,...,9,11} 1-tentactic 3-Sextactic 3 {1, 2, 3, ..., 7, 9, 10, 11} 2 {1,2,3,...,9,12} 2-tentactic 4-Sextactic 6 {1, 2, 3, ..., 7, 10, 11, 12} 3 {1,2,3,...,9,13} 3-tentactic 9 {1, 2, 3, ..., 7, 11, 12, 13} 5-Sextactic 4 {1,2,3,...,9,14} 4-tentactic 6-Sextactic 12 {1, 2, 3, ..., 7, 12, 13, 14} 5 {1,2,3,...,9,15} 5-tentactic 7-Sextactic 15 {1, 2, 3, ..., 7, 13, 14, 15} 6 {1,2,3,...,9,16} 6-tentactic 7 {1,2,3,...,9,17} 7-tentactic 8 {1,2,3,...,9,18} 8-tentactic
Proof. It follows, by Theorem 2.11 and Bezout’s theorem, that 6 6 Dp meets X at p with 8 ls 12. Hence, varying ls produces the last table. h Proof. Since the point p is neither flex nor sextactic, then
dim Q ‘ p 9 ‘ for ‘ 1; 2; 3; ...; 9: Corollary 2.13. If a smooth projective curve X of degree 6 has 1- ð Þ¼ ¼ ð1Þ sextactic or 2-sextactic points, then X is hyperelliptic. Hence, 1; 2; 3; ...; 9 2 Gp ðQÞ. Moreover, assuming that IðX; EpÞ¼lt, then Corollary 2.14. On a smooth non-hyperelliptic projective curve div ðE Þ2Qð l pÞ Qð ð1 þ l Þ pÞ: X of degree 6, the 1-weight of a sextactic point is given by p t t ð1Þ ð1Þ xp ðQÞ¼3ls 21, where ls is the multiplicity of the osculating Therefore, 1 þ lt 2 Gp ðQÞ. Consequently, GpðQÞ¼f1; 2; ð1Þ 6 h conic Dp at p. 3; ...; 9; 1 þ ltg. Finally, 19 R Gp ðQÞ as n10 18.
Proof. If p is a sextactic point on C, then Corollary 2.18. If a smooth projective curve X of degree 6 has 9- ð1Þ Gp ðQÞ¼f1; 2; 3; ...; 7; 1 þ ls; 2 þ ls; 3 þ lsg. Consequently, tentactic points, then X is hyperelliptic. ðqÞ ð1Þ g Notation. Ti ðCÞ will denote the set of i-tentactic points which q xp ðQÞ :¼ Rr¼0ðnr rÞ ð Þ are q-Weierstrass points on X and NTi ðXÞ will denote the ðqÞ ¼ð1 þ ls 8Þþð2 þ ls 9Þþð3 þ ls 10Þ cardinality of the set Ti ðXÞ. ¼ 3l 21: s Corollary 2.19. For a smooth projective non-hyperelliptic curve ð1Þ C of degree 6, the maximal cardinality of Ti ðXÞ is given by the ðqÞ Notation. Si ðXÞ will denote the set of i-sextactic points which following inequality ðqÞ are q-Weierstrass points on X and NSi ðXÞ will denote the car- dinality of the set SðqÞðXÞ. 990 i NTð1Þ 6 ; i i Corollary 2.15. For a smooth non-hyperelliptic projective ð1Þ where i ¼ 1; 2; 3; ...; 8 and ½m is the greatest integer less than or curve X of degree 6, the maximal cardinality of S ðXÞ is i equal to m. given by 990 NSð1ÞðXÞ 6 ; 3. Concluding remarks i 3ði þ 5Þ 21 where i ¼ 3; 4; 5; 6; 7 and ½m is the greatest integer less than or We conclude the paper by the following remarks and equal to m. comments.
In this article, the 1-gap sequence of the 1-Weierstrass 2.3. Tentactic points points on smooth non-hyperelliptic algebraic curves of degree 6 is computed. Furthermore, the geometry of Wei- Definition 2.16 (11,18). A point p on a smooth plane curve C of erstrass points is classified as flexes, sextactic and tentactic genus g P 2, which is neither flex nor sextactic point, is said to points. On the other hand, we show that a smooth non- be a tentactic point, if there a cubic Ep which meets C at p with hyperelliptic curve of degree 6 has no 4-flex, no 1-sextactic, contact order at least 10. The positive integer t such that no 2-sextactic and no 9-tentactic points. Also, an upper i :¼ IðC; Ep; pÞ 9 is called the tentactic order of p. Moreover, bound for the numbers of flexes, sextactic and tentactic the point p is said to be i-tentactic. points on such curves is estimated. The main theorems constitute a motivation to solve more Theorem 2.17. Let p be a tentactic point on a smooth projective general problems. One of these problems is the investigation non-hyperelliptic curve X of degree 6 and let Ep be its osculating of the geometry of higher order and multiple Weierstrass ð1Þ points on non-hyperelliptic degree 6 curves.Another cubic curve such that IðX; Ep; pÞ¼lt. Then Gp ðQÞ¼ problem is combining the main results together with the f1; 2; 3; ...; 9; 1 þ ltg. Moreover, the geometry of such points is given by the table: classification of degree 6 non-hyperelliptic curves in [19] Gap sequences of 1-Weierstrass points on non-hyperelliptic curves of genus 10 321
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