An Example of a Covering Construction in Complex Projective Space

In algebraic geometry as in many fields of mathematics, one tries to classify the objects studied. The objects in algebraic geometry are called varieties. An important tool in classification theory is the construction of new varieties starting from well-known varieties.

Using the (affine) cone over a smooth projective variety we construct a new variety. We show that we obtain a covering morphism between algebraic varieties which is branched over some divisor. We mainly use some Bertini type theorem. Then the theory of branched coverings can be used to determine the topological invariants of the constructed (new) variety. E.g. we compute its sectional genus.