Consider the irreducible complex polynomial p E C[r, y] given by p(x, y) = y +...

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Consider the irreducible complex polynomial p E C[r, y] given by p(x, y) = y² + 1. Let Σ := p=1(0) C C2 and let Σ > Σ be a compactification of E, as in Lecture 21. 3.1. Prove that is a smooth 2-manifold. It suffices to prove that 0 is a regular value of p. 3.2. For the map : E→C, (x,y) →x, determine the branch points ; of and the indicies np(x). Justify your answers. 3.3. Using the Riemann-Hurwitz Formula, determine x(E), the Euler characteristic of Σ. 3.4. Determinen, (oo), the index at infinity, and so determine x() and hence the topological type of E. Justify your answers. Consider the irreducible complex polynomial p E C[r, y] given by p(x, y) = y² + 1. Let Σ := p=1(0) C C2 and let Σ > Σ be a compactification of E, as in Lecture 21. 3.1. Prove that is a smooth 2-manifold. It suffices to prove that 0 is a regular value of p. 3.2. For the map : E→C, (x,y) →x, determine the branch points ; of and the indicies np(x). Justify your answers. 3.3. Using the Riemann-Hurwitz Formula, determine x(E), the Euler characteristic of Σ. 3.4. Determinen, (oo), the index at infinity, and so determine x() and hence the topological type of E. Justify your answers.

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31 To prove that is a smooth 2manifold we need to show ... View the full answer