Pencil Of Surfaces Research Articles

Twistors, Quartics, and del Pezzo Fibrations

It has been known that twistor spaces associated to self-dual metrics on compact 4-manifolds are source of interesting examples of non-projective Moishezon threefolds. In this paper we investigate the structure of a variety of new Moishezon twistor spaces. The anti-canonical line bundle on any twistor space admits a canonical half, and we analyze the structure of twistor spaces by using the pluri-half-anti-canonical map from the twistor spaces. Specifically, each of the present twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of rational curves, and it is an anti-canonical curve on smooth members of the pencil. These twistor spaces are naturally classified into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic hypersurface has to be of a specific form. Together with our previous result in Honda (“A new series of compact minitwistor spaces and Moishezon twistor spaces over them”, 2010), the present result completes a classification of Moishezon twistor spaces whose half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is smaller than a pencil. Twistor spaces which have a similar structure were studied in Honda (“Double solid twistor spaces: the case of arbitrary signature”, 2008 and “Double solid twistor spaces II: General case”, 2015) and they are very special examples among the present twistor spaces.

A pencil of Enriques surfaces with non-algebraic integral Hodge classes

We prove that there exists a pencil of Enriques surfaces defined over $\mathbb{Q}$ with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three.

Critical Loci for Higgs Bundles

The paper studies the locus in the moduli space of rank 2 Higgs bundles over a curve of genus g corresponding to points which are critical for d of the Poisson commuting functions defining the integrable system. These correspond to the Higgs field vanishing on a divisor D of degree d. The degree d critical locus is shown to have an induced integrable system related to K(−D)-twisted Higgs bundles. It is embedded in the singular part of the fibration and a description of these singular fibres using Hecke curves is given. The methods used are also applied to give information about the cohomology classes of components of the nilpotent cone and of the first critical locus. It is shown that whereas in the extreme case d = 2g − 2 the locus is a hyperkähler submanifold this does not hold in general. The example of genus 2 is studied concretely and the d = 1 integrable system is seen to be described by a pencil of Kummer surfaces.

Derived equivalent Calabi–Yau threefolds from cubic fourfolds

We describe pretty examples of derived equivalences and autoequivalences of Calabi-Yau threefolds arising from pencils of cubic fourfolds. The cubic fourfolds are chosen to be special, so they each have an associated K3 surface. Thus a pencil gives rise to two different Calabi-Yau threefolds: the associated pencil of K3 surfaces, and the baselocus of the original pencil—the intersection of two cubic fourfolds. They both have crepant resolutions which are derived equivalent.

Noncommutative quadric surfaces

The 4-dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3-space. The degree-two component of the algebra contains a 2-dimensional subspace of central elements. The zero loci of those central elements, except 0, form a pencil of noncommutative quadric surfaces. We show that the behavior of this pencil is similar to that of a generic pencil of quadrics in the commutative projective 3-space. There are exactly four singular quadrics in the pencil. The singular and non-singular quadrics are characterized by whether they have one or two rulings by noncommutative lines. The Picard groups of the smooth quadrics are free abelian of rank two. The alternating sum of dimensions of Ext groups allows us to define an intersection pairing on the Picard group of the smooth noncommutative quadrics. A surprise is that a smooth noncommutative quadric can sometimes contain a “curve” having self-intersection number −2. Many of the methods used in our paper are noncommutative versions of methods developed by Buchweitz, Eisenbud and Herzog: in particular, the correspondence between the geometry of a quadric hypersurface and maximal Cohen–Macaulay modules over its homogeneous coordinate ring plays a key role. An important aspect of our work is to introduce definitions of noncommutative analogues of the familiar commutative terms used in this abstract. We expect the ideas we develop here for 2-dimensional noncommutative quadric hypersurfaces will apply to higher dimensional noncommutative quadric hypersurfaces and we develop them in sufficient generality to make such applications possible.

A pencil of Enriques surfaces of index one with no section

Monodromy arguments and deformation-and-specialization are used to prove existence of a pencil of Enriques surfaces with no section and index 1. The same technique completes the strategy from [GHMS05, §7.3] proving the family of witness curves for dimension d depends on the integer d.

Special members in the bicanonical pencil of Godeaux surfaces

The object of this paper is to find the number of hyperelliptic curves in the bicanonical pencil of a Godeaux surface whose torsion group is Z3, or Z4, or Z5. Let be a minimal smooth projective surface of general type with = = 0, 2 = 1. Surfaces of this type are called Godeaux surfaces (also they are called numerical Godeaux surfaces ); they were discovered in the 1930s by Campedelli [2] and Godeaux [6], and are interesting in view of Castelnuovo’ s criterion: an irrational surface with = 0 must have 2 1. The following well known results are important tools in the paper: Bicanonical system 2 gives a pencil and the fixed part of 2 cons ists of 2-curves [12]; Tricanonical system 3 gives a birational map P 3 without fixed components [3], [12]; Denote by T( ) the torsion subgroup of H 2 ( Z). Recall that 0, Z2, Z3, Z4, Z5 are the only possible values for the torsion group of a Godeaux surface [12], [13]; A half of the number of elements in the set T( ) = equals the number of base points of 3 [12]; and For each non-zero element T( ), 0 ( + ) = 1. And + 1 intersects + 2 transversally if 1 = 2 [13].

Some Calabi–Yau threefolds with obstructed deformations over the Witt vectors

Some smooth Calabi–Yau threefolds in characteristic two and three that do not lift to characteristic zero are constructed. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian surfaces and Katsura's analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.

Integrable mappings via rational elliptic surfaces

We present a geometric description of the QRT map (which is an integrable mapping introduced by Quispel, Roberts and Thompson) in terms of the addition formula of a rational elliptic surface. By this formulation, we classify all the cases when the QRT map is periodic; and show that its period is 2, 3, 4, 5 or 6. A generalization of the QRT map which acts birationally on a pencil of K3 surfaces, or Calabi–Yau manifolds, is also presented.

A Second-Order Linear System with Fuchsian Singularities on CPm

here Uj is an unknown second-order square matrix, and the Pj(x) are irreducible homogeneous polynomials of degree pj in x = (x1, x2, . . . , xm+1) ∈ CP. Suppose that system (1) has no Fuchsian singularities at infinity. Then the differential matrices Uj of system (1) must satisfy the condition ∑n j=1 pjUj = 0. By [1, 2], Eq. (1) is completely integrable if and only if the differential form (2) satisfies the condition ω ∧ ω = 0, (3) where ∧ is the operator of exterior product of differential forms. It was shown in [3] that conditions (3) are necessarily satisfied if the manifolds Mj, j = 1, . . . , n, form a pencil of algebraic surfaces. Then there exist two polynomials Q and R such that Pj = αjQ+ βjR for all j = 1, . . . , n, where αj , βj ∈ C, and Eq. (1) can be reduced to the onedimensional system

Vector Bundles on Del-Pezzo Fibrations

We study stable rank two bundles on a threefold with a pencil of Del-Pezzo surfaces. When the generic fiber is a cubic surface in P3, we give a description of the moduli spaces of such bundles.

Birational properties of pencils of del Pezzo surfaces of degrees 1 and 2

The problem of birational rigidity is investigated for smooth Mori fibre spaces representable as a pencil of del Pezzo surfaces of degree 1 or 2.

Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces

In this paper it is proved that there is only one pencil of rational surfaces on a smooth three-dimensional variety fibred into Del Pezzo surfaces of degree 1, 2 or 3 by a morphism (Iskovskikh's conjecture), provided that the class of 1-cycles (where is the canonical class and is the class of a line in a fibre) is not effective for any . This condition is satisfied if is sufficiently “twisted” over the base of the pencil. This implies that these varieties admit no conic bundle structures, and, in particular, that they are nonrational. Certain higher-dimensional generalizations of this theorem are considered: similar statements are true for varieties with a pencil of thee-dimensional quartics of general position, and analogous results are obtained in these cases.

Classification of Twistor Spaces with a Pencil of Surfaces of Degree 1, Part I

AbstractWe display a family of compact complex 3‐manifolds which yields all twistor spaces containing a pencil of surfaces transversal to the twistor lines. This also yields explicit families of self‐dual metrics on connected sums of complex projective planes. This yields a completely alternative approach to a class of such metrics which was recently found by Claude le Brun [11].

Intermediate Jacobians and Chow groups of threefolds with a pencil of del Pezzo surfaces

In this paper we study threefolds with a pencil of del Pezzo surfaces with 1⩽K2⩽5. The curve which parametrises the exceptional curves on the fibers and the incidence correspondence on it determine a Prym-Tjurin variety which is an abelian subvariety of the Jacobian of the curve. Under certain restrictions on the singular fibers of the pencil we prove that there is a polarized isomorphism between the intermediate Jacobian of the threefold and the Prym-Tjurin variety and an isomorphism between the latter and the Chow group of algebraic one-cycles algebraically equivalent to zero.

Binary forms and pencils of quadrics

The incitement, or provocation, to assemble the following pages was the recognition in a recent paper (5) of a geometrical property characterizing those pairs, in a pencil of quadric surfaces, such that the quadrics of a pair are both outpolar and inpolar to each other: symmetrically apolar let it be said. That there are three such pairs was known (8), but their identification by means of the principal chords of their common curve had apparently not been remarked upon. The question naturally arose as to the number and identification of such pairs of quadrics of a pencil in n-dimensional projective space [n]; it is easy to show that the number is ½n(n–1), but geometrical identification seems quite another matter and no success whatever has yet been achieved even when n is 4.

On the quartic combinant of a pencil of quadric surfaces

Un covariante biquadratico di un fascio di quadriche inS3, gia ottenuto analiticamente dall'autore, vien qui discusso ed approfondito dal punto di vista geometrico.

Combinants of a pencil of quadric surfaces (IV)

In this paper we determine the complete system of combinantal line-complexes associated with a pencil of quadric surfaces. The method employed is that already used in (I) and (II) to obtain the complete systems of combinantal covariants and contravariants.

Combinants of a pencil of quadric surfaces (III)

The two previous papers of this series dealt with the determination of the complete system of combinantal covariants and contravariants of a pencil of quadric surfaces. The present paper is concerned with combinantal forms involving line-complexes, and with certain syzygies which exist between these forms. It is essentially a preparation for the determination of the complete system of combinantal line-complexes of the pencil, which will be carried out explicitly in the next paper of the series.

Combinants of a pencil of quadric surfaces. I

The classical account of the invariants and covariants of a pair of quadric surfaces is due to Salmon. The actual determination, in explicit form, of the complete system of concomitants of two quaternary quadratic forms is much later in date, and is due to Turnbull. This system, which includes mixed concomitants of various kinds, is complicated. As originally determined, it comprised 125 forms, three of which were later shown to be reducible. In the accounts of both these authors, however, the forms are considered as belonging to a particular pair of quadric surfaces, and the problem of determining these covariant forms which are invariant, not merely under change of coordinate system but also under change of base in the pencil denned by the two quadrics, is not alluded to. Such forms, which are of obvious geometrical interest, are called combinants. It is rather surprising to find that very little seems to be known about the combinants attaching to a pencil of quadric surfaces; the Encyklopädie scarcely mentions them, and I have been unable to trace any references in the literature except to the two invariants whose vanishing expresses the condition that the parameters of the four cones in the pencil form an equianharmonic or a harmonic set.