Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 146 / 4 / c

Solution

Let

F_{0}

and

F_{1}

be general homogeneous cubic forms with base locus

B

of nine points. The incidence variety

E (1) = {([z], [s : t]) \in CP^{2} \times CP^{1} : s F_{0} (z) + t F_{1} (z) = 0}

is the blowup of

CP^{2}

B

π : E (1) ⟶ CP^{1}, ([z], [s : t]) ⟼ [s : t],

and its fibers are connected plane cubics. Over any base point

p \in B

, the exceptional curve

E_{p} = {p} \times CP^{1}

maps isomorphically to the base, so it is a holomorphic section.

Write

H

for the pullback of a line and

E_{1}, \dots, E_{9}

for the exceptional classes. The fiber class and the canonical class of the rational elliptic surface are

F = 3 H - \sum_{i = 1}^{9} E_{i}, K_{E (1)} = - 3 H + \sum_{i = 1}^{9} E_{i} = - F .

The degree of

π ∣_{C}

C \cdot F = d

2 g (C) - 2 = C^{2} + K_{E (1)} \cdot C = k - d .

g (C) = 1 + \frac{k - d}{2} .

There are no discussions about this article yet.

There are currently no matching articles.