By David Bressoud

This booklet is an undergraduate creation to actual research. lecturers can use it as a textbook for an leading edge path, or as a source for a normal direction. scholars who've been via a standard path, yet don't realize what genuine research is set and why it was once created, will locate solutions to a lot of their questions during this booklet. even if this isn't a heritage of study, the writer returns to the roots of the topic to make it extra understandable. The publication starts off with Fourier's advent of trigonometric sequence and the issues they created for the mathematicians of the early 19th century. Cauchy's makes an attempt to set up an organization origin for calculus stick with, and the writer considers his disasters and his successes. The ebook culminates with Dirichlet's facts of the validity of the Fourier sequence growth and explores a number of the counterintuitive effects Riemann and Weierstrass have been resulted in due to Dirichlet's facts. Mathematica ® instructions and courses are incorporated within the routines. even if, the reader could use any mathematical device that has graphing services, together with the graphing calculator.

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**Example text**

We next ask more about the curve itself. After the genus, the gonality and the Clifford index are among the most interesting invariants. Recall that the gonality of X is the smallest degree of a mapping from X to lP'1. To define the Clifford index of X we first define the Clifford index of a line bundle L on X to be degree(L) - 2(hO(L) - 1). For example, the Clifford indices of the structure sheaf Ox and the canonical sheaf Wx are both equal to O. The Clifford index of the curve X is defined to be the minimum value of the Clifford index of a line bundle L on X for which both hO(L) ;::: 2 and h I (L) ;::: 2.

Next, the cohomology class of X in jp3 is determined by the degree of X: i63 : degree idealX 063 = 10 In sum: X is a smooth, absolutely irreducible curve of genus 6 and degree 10. We next ask for analytic information about the curve and the embedding. A reasonable place to start is with the relation between the line bundle defining the embedding and the canonical sheaf wx. Notice first that the degree of the hyperplane divisor (the degree of the curve) is 10 = 2g-2, the same as the canonical bundle.

I53 : X = variety idealX 053 = X 053 : Projective Variety How would you analyze the scheme X? We will illustrate one approach. In outline, we will first look at the topological invariants: the number and dimensions of the irreducible components, and how they meet if there is more than one; the topological type of each component; and the degree of each component in jp>3. We will then see what we can say about the analytic invariants of X using adjunction theory (we give some references at the end).