Spanned Line Bundle Research Articles

Comment on: On the irreducibility of the Severi variety of nodal curves in a smooth surface, by E. Ballico

In this short note, I point out that results of Ballico and Kool–Shende–Thomas together imply that on K3, Enriques, and Abelian surfaces, if L is a very ample and $$(2p_a(L)-2g-1)$$-spanned line bundle, then the equigeneric Severi variety $$V_{g}(L)$$ of all curves in |L| having genus g is non-empty, irreducible, of the expected dimension, and its general member is a $$(p_a(L)-g)$$-nodal curve.

Classification of polarized manifolds by the second sectional Betti number

Let X be an n-dimensional smooth projective variety defined over the field of complex numbers, let L be an ample and spanned line bundle on X. Then we classify (X,L) with b2(X,L) = h2(X,ℂ)+1, where b2(X,L) is the second sectional Betti number of (X,L).

THE LÜROTH SEMIGROUP OF A PLANE CURVE C OVER 𝔽qAND THE PRIMITIVE SET OF P ∈ C(𝔽q)

Let C ⊂ ℙ2be a smooth and geometrically connected curve defined over 𝔽q. We study the Lüroth semigroup of C over 𝔽q, i.e. the set L(C, q) of all degrees > 0 of some spanned line bundle on C defined over 𝔽q. Fix P ∈ C(𝔽q). We also study the set of all t ∈ ℕ such that both [Formula: see text] and ωC(–tP) are spanned (the primitive set of P), when q is a square, C is the Hermitian curve and P ∈ C(𝔽q).

The Lüroth semigroups of a curve over a non-algebraically closed field

Let C⊂P2 be a smooth curve defined over a non-algebraically closed field K. We study the Lüroth semigroups of C over K, i.e. the set L′(C,K) of all degrees of finite morphisms C→P1 defined over K and the set L(C,K) of all degrees >0 of some spanned line bundle on C defined over K. If K is infinite, then L′(C,K)=L(C,K), but for every prime power q≠2 there is a smooth plane curve C defined over Fq with L′(C,Fq)⊆L(C,Fq) and C(Fq)≠∅. If C is a smooth plane curve, then L(C,K) determines (in several ways) if C(K)≠∅.

Discriminant loci of ample and spanned line bundles

Let ( X , L , V ) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V ⊆ H 0 ( X , L ) spans L . The discriminant locus D ( X , V ) ⊂ | V | is the algebraic subset of singular elements of | V | . We study the components of D ( X , V ) in connection with the jumping sets of ( X , V ) , generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of ( X , L , V ) ) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.

Fano manifolds of coindex four as ample sections

Abstract We give a classification of pairs (X, L) consisting of a smooth complex n-dimensional projective variety X and an ample line bundle L on X such that Z ∈ is a Fano manifold with − K Z = (dim Z − 3)H Z for an ample and spanned line bundle H Z on Z.

Varieties with small discriminant variety

Let X be a smooth complex projective variety, let L be an ample and spanned line bundle on X, V C H°(X,L) defining a morphism Φ V : X → P N and let D(X,V) be its discriminant locus, the variety parameterizing the singular elements of |V|. We present two bounds on the dimension of D(X, V) and its main component relying on the geometry of Φ V (X) ⊂ P N . Classification results for triplets (X, L, V) reaching the bounds as well as significant examples are provided.

Bad loci of free linear systems

Abstract The bad locus of a free linear system ℒ on a normal complex projective variety X is defined as the set B( ℒ ) ⊆ X of points that are not contained in any irreducible and reduced member of ℒ. In this paper we provide a geometric description of such locus in terms of the morphism defined by ℒ. In particular, assume that dim X ≥ 2 and ℒ is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(ℒ) is empty unless X is a surface. Then we prove that, when the latter occurs, B(ℒ) is not empty if and only if ℒ defines a morphism onto a two dimensional cone, in which case B(ℒ) is the inverse image of the vertex of the cone.

Non-special projectively normal line bundles on general k-gonal curves

Let X be a sucien tly general smooth k-gonal curve of genus g and R2 Pic(X) the degree k spanned line bundle. We nd an optimal integer z > 0 such that the line bundle R z is very ample and projectively normal.

Trigonal gorenstein curves and special linear systems

LetY be a Gorenstein trigonal curve withg:=pa(Y)≥0. Here we study the theory of special linear systems onY, extending the classical case of a smoothY given by Maroni in 1946. As in the classical case, to study it we use the minimal degree surface scroll containing the canonical model ofY. The answer is different if the degree 3 pencil onY is associated to a line bundle or not. We also give the easier case of special linear series on hyperelliptic curves. The unique hyperelliptic curve of genusg which is not Gorenstein has no special spanned line bundle.

On higher order embeddings of Calabi-Yau threefolds

Let X be a smooth Calabi-Yau threefold and let L be an ample and spanned line bundle on X. We give lower bounds for nL to be k-jet ample or k-jet spanned.

Discriminant loci of varieties with smooth normalization

Let X be a smooth complex projective n-fold endowed with an ample and spanned line bundle (L). Under the assumption that Γ(L) defines a generically one-to-one map we describe the singular set of the general element in the main component of the discriminant locus of |L|. This description is used to show that (X:,L) is covered by linear Pk’s, where k + 1 stands for the codimension of the main component. We also give some applications relating k to the spectral value of (X, L) and discuss some examples.

Complex manifolds polarized by an ample and spanned line bundle of sectional genus three

Let X be a complex projective manifold with \( \dim X\geq 3 \) and let L be an ample and spanned line bundle on X. We classify polarized manifolds (X,L) of sectional genus three mainly by adjunction theory and the classification theory of \(\Delta \)-genus.

On the discriminant locus of an ample and spanned line bundle.

On the discriminant locus of an ample and spanned line bundle.

Complex surfaces polarized by an ample and spanned line bundle of genus three

Let X be a complex connected projective nonsingular algebraic surface endowed with an ample line bundle L, which is spanned by its global sections. Pairs (X, L) as above, with sectional genus g(X, L)=1+(L·(K X ⊗L))/2=3 are classified by means of the main techniques of adjunction theory.

On complex projective surfaces with trigonal hyperplane sections

Let S be a complex projective surface endowed with an ample and spanned line bundle L. Assume that (S,L) does not belong to some special classes and that cl(L)2≥10. We prove that(KS⊗L)·KS≤−3 and |L| contains a trigonal curve (of genus≥4) iff either (S,L) is a rational surface ruled by cubics, or the g13 of C is cut out by |KS⊗−1|. This result applies to surface having a hyperplane section which is a trigonal curve.