| Co-Investigator(Kenkyū-buntansha) |
HOMMA Masaaki Kanagawa U., Faculty of Engineering, Prof., 工学部, 教授 (80145523)
KATO Takao Yamaguchi U., Faculty of Science, Prof., 理学部, 教授 (10016157)
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| Research Abstract |
First, we have some result about normal generation of a line bundle L on a curve C (T.Kato, C.Keem and Ohbuchi), i.e. Green-Lazarsfeld says that if degL 【greater than or equal】 2g+2-Cliff (C), then L is normally generated, equivalently if L does not contribute to Clifford index, then L is normally generated. We get some sufficient condition of normal generation for a line bundle which contributes to a Clifford index. Next, we classify smooth projective algebraic curves C of genus g such that the variety of special linear systems W^2_<g-1> (C) has dimension g-7. We can prove that if W^2_<g-1> (C) has dimension g-7 【greater than or equal】 0 then C is either trigonal, tetragonal, a double covering of a curve of genus 2 or a smooth plane sextic. This result is an extension of H.Martens and D.Mumford Theorem. And we can classify curves in which W^2_<g-1> (C) has dimension g-7. The result is that dimW^2_<g-1> (C)=g-7 is equivalent to the following conditions according to the values of the ge
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nus g.
(i) C is either a trigonal curve or a double covering of a curve of genus two for g【greater than or equal】11.
(ii) C is either trigonal, a double covering of a curve of genus 2 or a smooth plane curve degree 6 for g=10.
(iii) C is either trigonal, a double covering of a curve of genus 2, a tetragonal curve with a smooth model of degree 8 in P^3 or a tetragonal curve with a plane model of degree 6 for g=9.
(iv) C is either trigonal or has a birationally very ample g^2_6 for g=8 or g=7.
And furthermore we have some result about Castelnuovo-Mumford type inequality. We consider m (L) : =min{m|h^1(X, L^<(【cross product】)m>)=0}. It is easy that m(L)【less than or equal】[(d -1)/(r-1)], and we can conclude that m(L)【less than or equal】d-r for any birationally very ample invertible sheaf L, moreover equality holds if and only if the curve is a smooth plane curve and L=O(1). From this result, we can consider an invariant θ(L) : =deg L-r(L)-m(L) for any birationally very ample invertible sheaf L.We can classift a curve C and a line bundle L which satisfies θ(L)=1. Less
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