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3-D Shapes Are Like Green Grapes! by Tracy Kompelien

By Tracy Kompelien

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Over a field K of characteristic y~, ... ,y:. of K such that 30 L. CHIANTINI, F. CIOFFI AND F. 1 ) where A 1= 0, 1, see (Silverman, III, Prop. 7). 1) is a Weierstrass representation of an elliptic plane curve in the Legendre form. The condition A 1= 0, 1 is equivalent to the fact that E is smooth, see for example (Silverman and Tate, IV, section 3). We suppose that A E K. Every field K contains a copy of Q or of Z p for some prime p and so we can assume that K = Q or Z p' To study an algebraic object over Q it is useful to study the reduction of the object modulo primes.

Hence, the maximal rank and the minimal generation of such curves are studied. In particular, we consider elliptic curves and we describe a method to compute their Hilbert functions in any dimension and for every choice of the degrees. Applications to the study of elliptic curves on threefolds are shown. Key words: Elliptic curves, Hilbert function. Mathematics Subject Classification (2000): 14H52, 14Q05. 1. Introduction In this paper we study the maximal rank and the minimal generation of disjoint unions of distinct non special curves, with applications to elliptic curves.

2. The Segre Conjecture ,2 Let PI, ... , Pn be general points in the complex projective plane and let m I, ... , mn be positive integers. We let ~J be the linear system of plane curves of degree d and ~ (p';ll .... , p;;lll) the sub-linear system of ~ of curves having multiplicity at least mj at the point Pi, i = 1, ... , n. ( /111 11111) Pi! ~PI ""'P/1' Let 1t : S -+ JlD2 be the blow-up of JlD2 at the points PI, ... ,Pn. Let £ be a line bundle on S, or, by abusing notation, the corresponding complete linear system.

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