By J. Rafael Sendra

The primary challenge thought of during this booklet is the decision of rational parametrizability of an algebraic curve, and, within the optimistic case, the computation of an excellent rational parametrization. This quantities to opting for the genus of a curve, i.e. its whole singularity constitution, computing ordinary issues of the curve in small coordinate fields, and developing linear structures of curves with prescribed intersection multiplicities. a number of optimality standards for rational parametrizations of algebraic curves are mentioned. This ebook is principally meant for graduate scholars and researchers in confident algebraic curve geometry.

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If the dimension is one, the linear system is also called a pencil of curves. An interesting type of linear systems arises when we require the curves to pass through given points with given multiplicities. This motivates the following deﬁnition. 54. P ∈ P2 (K) is a base point of multiplicity r ∈ N of a linear system H of curves of ﬁxed degree, if every curve C in H satisﬁes multP (C) ≥ r. 55. We deﬁne the linear system of curves of degree d generated by the eﬀective divisor D = r1 P1 + · · · + rm Pm as the set of all curves C of degree d such that multPi (C) ≥ ri , for i = 1, .

50, multO (e, y) is equal to multO (x4 , y), which in turn is equal to 4 by relations (6), (5). By relation (5), multO (e, h) is equal to multO (e) · multO (h), which is 6. Thus, multO (E, F) = multO (e, f ) = 14. 4 Linear Systems of Curves Linear systems of curves are an indispensably tool in algebraic geometry. In this section we derive some basic properties of linear systems of curves, based on the exposition in [Mir99]. The idea of linear systems of curves is to work with sets of curves of ﬁxed degree related by means of some linear conditions; for instance, sets of curves of ﬁxed degree passing through some speciﬁc points with at least some ﬁxed multiplicities.

Substituting a nonzero power series of positive order into the coordinates of a local parametrization yields a parametrization with the same center. 72. Two (aﬃne or projective) local parametrizations P1 (t), P2 (t) of an algebraic curve C are called equivalent iﬀ there exists A ∈ K[[t]] with ord(A) = 1 such that P1 = P2 (A). 68 we see that this equivalence of local parametrizations is actually an equivalence relation. 73. In a suitable aﬃne coordinate system any given local parametrization is equivalent to one of the type tn , a1 tn1 + a2 tn2 + a3 tn3 + · · · , where 0 < n, and 0 < n1 < n2 < n3 < · · ·.