On non-projective normal surfaces

Abstract

In this note we construct examples of non-projective normal proper algebraic surfaces and discuss the somewhat pathological behaviour of their Neron-Severi group. Our surfaces are birational to the product of a projective line and a curve of higher genus.

On embeddings into toric prevarieties

“…Surfaces as above really exist; compare [9], 2.5. Here the assumption that the ground field k is uncountable comes in.…”

There are enough Azumaya algebras on surfaces

“…Remark 4.3. It might easily happen that X has trivial Picard group [25]. However, the preceding results ensures the existence of vector bundles of higher rank.…”

A higher-dimensional generalization of Mumford's rational pullback for Weil divisors

“…According to [11], Proposition 1.6, there is an effective Cartier divisor Z ⊂ Y ⊗ A k so that on the blowing-up ϕ : Bl Z (Y ) → Y , the strict transform E of the closed fiber Y ⊗ A k admits a contraction ϕ ′ : Bl Z (Y ) → Y ′ to some projective A-scheme Y ′ . Such a construction resembles the elementary transformations for projective bundles, and was already used in [10] for surfaces. Using Bertini, one can arrange things that Z is smooth, hence the total space Bl Z (Y ) is regular.…”