Homework for Week 9

Part 1

The Segre embedding f: P¹ x P² → P⁵ is defined by f(x₀, x₁, y₀, y₁, y₂) = [x₀ y₀, x₀ y₁, x₀ y₂, x₁ y₀, x₁ y₁, x₁ y₂]. Show that its image is a projective variety with no singularities. Compute the dimension of its image. Compute its degree. In general, the Segre embedding f: P^r x P^s → P^rs+r+s can be defined as the products [..., x_i y_j, ...] in lexicographic order. Its image is always a nonsingular projective variety. Can you conjecture/compute/prove something about its dimension? What about its degree?

Part 2

1. Determine the divisor of x/y on the quadric surface xy-zw=0 in P³.
2. Determine the divisor of the function (x/y)-1 on the circle x² + y² = z² in P².

• Part 1: Due Tuesday
• Part 2: Due Thursday