Rational conic fibrations of sectional genus two

Polarized rational surfaces (X, L) of sectional genus two ruled in conics are studied. When they are not minimal, they are described as the blow-up of 𝔽1 at some points lying on distinct fibers. Ampleness and very ampleness of L are studied in terms of their location. When L is very ample and there is a line contained in X and transverse to the fibers, the conic fibrations (X, L) are classified and a related property concerned with the inflectional locus is discussed.

Asymptotic Syzygies and Higher Order Embeddings

We show that vanishing of asymptotic $p$-th syzygies implies $p$-very ampleness for line bundles on arbitrary projective schemes. For smooth surfaces, we prove that the converse holds when $p$ is small, by studying the Bridgeland–King–Reid–Haiman correspondence for tautological bundles on the Hilbert scheme of points. This extends the previous results by Ein–Lazarsfeld and Ein–Lazarsfeld–Yang and gives a partial answer to some of their questions. As an application of our results, we show how to use syzygies to bound the irrationality of a variety.

NOTES ON TORIC VARIETIES FROM MORI THEORETIC VIEWPOINT, II

We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita’s freeness and very ampleness for toric varieties.

Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients

The purpose of this note is to show that{2K}of any smooth compact complex 2-ball quotient is very ample, except possibly for four pairs of fake projective planes of minimal type, whereKis the canonical line bundle. For the four pairs of fake projective planes, the sections of{2K_{M}}give an embedding ofMexcept possibly for at most two points onM.

Criteria for Very Ampleness of Rank Two Vector Bundles over Ruled Surfaces

Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.