On the Luroth semigroup of smooth Plane curves

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In this project, we have studied base point free pencils on a triple coverin of an algebraic curve of positive genus. Let X be a smooth algebraic curve of genus g which admits a triple sheeted covering onto a general curve C (in the sense of Brill-Noether) of genus h <greater than or equal> 1. Letpi : X*C be the covering projection. By a simple application of the Castelnuovo-Severi inequality one can easily see that there does not exist a base point free pencil of degree less than or equal to (g-3h)/ on X other than the pull-backs from the base curve C.Hence, a relatively low part of the Luroth semigroup of X is determined that of C,completely. So it is important to seek the degrees of pencils on X which are not composed with pi. The reason why we studied the case h > 0 and triple sheeted is the following :i) For the case h = 0 and two sheeted, X is hyperelliptic, so the Luroth semigroup is well-known.ii) For the case h = 0 and three sheeted, it was already solved by G.Martens and F.-O … More .Schreyer.iii) For the case h = 0 and n (<greater than or equal> 4)-sheeted, recently, Coppens-Keem-Martens solved for generalThe main theorem we obtained is the following :THEOREM.Let X and C be as above. If g <greater than or equal> (2 [(3h+1)/] +1) ([(3h+1)/] +1), then there exists a base point free pencil of any degree d <greater than or equal> g-[(3h+1)/] -1 on X which is not composed with pi.For d <greater than or equal> g-[(3h+1)/], applying the enumerative calculus for the intersection theory to subvarieties W^r_ (X) in the Jacobian variety J (X), we can prove our theorem not so hard. However, for d = g-[(3h+1)/] -1, it was hard to prove Theorem bacause it was difficult to prove the irreducibility of W^1_. Finaly, we succeeded it by use of the results and discussion of R.Miranda's paper appearing in Amer.J.Math.107 (1985).It is noted that the bound of degree in our theorem is far from the Castelnuovo-Severi bound. So it seems not to be the best. In fact, observing cyclic triple covering, we have :i) In general, Castelnuovo-Severi bound is the best. But it can be improved for special cases.ii) For case of cyclic triple coverings, we will be able to improve our theorem substantially. Less