Titlebook: Algebraic Geometry; An Introduction Daniel Perrin Textbook 2008 Springer-Verlag London 2008 Algebra.Arithmetic.Riemann-Roch theorem.algebra

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https://doi.org/10.1007/978-1-4419-5548-7Our aim is to show that two plane curves of degrees . and . have exactly . intersection points. We saw in the introduction that we need to take care when stating this result.

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https://doi.org/10.1007/978-3-319-42076-9We assume that the field . is algebraically closed. We will use the following notation:

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Affine algebraic sets,Let . be a positive integer. Consider the space.. If . = (., …, .) is a point in . and .(., …, .) is a polynomial, we denote .(., …, .) by .(.). The first fundamental objects we encounter are the affine algebraic sets defined below.

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Liaison of space curves,We assume that the field . is algebraically closed. We will use the following notation:

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Projective algebraic sets,n intersections always contain a certain number of special cases due to parallel lines or asymptotes. For example, in the plane two distinct lines meet at a unique point except when they are parallel. In projective space, there are no such exceptions.