Fano 3-folds Research Articles

Moduli of certain Fano 4-folds

In this brief note we give a proof that a certain family of Fano 4-folds described below is complex locally complete and effectively parametrized in the sense of Kodaira-Spencer [Ko-Sp].

Deformations of weak Fano 3-folds with only terminal singularities

In this article, we prove that any $\Bbb Q$-factorial weak Fano 3-fold with only terminal singularities has a smoothing.

Fano Hypersurfaces in Weighted Projective 4-Spaces

We determine the full list of anticanonically embedded quasismooth Fano hypersurfaces in weighted projective 4-spaces. There are 48 infinite series and 4442 sporadic examples. In particular, the Reid-Fletcher list of 95 types of anticanonically embedded quasismooth terminal Fano threefolds in weighted projective 4-spaces is complete. We also prove that many of these Fano hypersurfaces admit a Kähler-Einstein metric, and study the nonexistence of tigers on these Fano 3-folds. Finally, we prove that there are only finitely many families of quasismooth Calabi-Yau hypersurfaces in weighted projective spaces of any given dimension. This implies finiteness for various families of general type hypersurfaces.

Geometry and Algebra of Prime Fano 3-folds of Genus 12

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. The connection between these Fano 3-folds and plane quartic curves is explained.

Boundedness of canonical $\mathbf {Q}$-Fano 3-folds

We give an effective bound of the Gorenstein index of weak $\mathbf{Q}$-Fano 3-folds, and prove the boundedness of the terminal $\mathbf{Q}$-Fano 3-folds. Combined with [Bor99], this result implies furthermore the boundedness of the canonical $\mathbf{Q}$-Fano 3-folds.

Vector bundles on fano 3-folds without intermediate cohomology

A well known result of G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964; Zbl 0126.16801)] says that a vector bundle on a projective space has no intermediate cohomology if and only if it decomposes as a direct sum of line bundles. It is also known that only on projective spaces and quadrics there is, up to a twist by a line bundle, a finite number of indecomposable vector bundles with no intermediate cohomology [see R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165-182 (1987; Zbl 0617.14034) and also H. Kn¨orrer, Invent. Math. 88, 153-164 (1987; Zbl 0617.14033)]. In the paper under review the authors deal with vector bundles without intermediate cohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-folds they consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5 and smooth 3-dimensional linear sections of G(1, 4) P9. A complete classification of rank two vector bundles without intermediate cohomology on such 3-folds is given. In fact the authors prove that, up to a twist, there are only three indecomposable vector bundles without intermediate cohomology. Vector bundles of rank greater than two are also considered. Under an additional technical condition, the authors characterize the possible Chern classes of such vector bundles without intermediate cohomology.

Action of Cyclic Groups on Fano 3-Folds

Action of Cyclic Groups on Fano 3-Folds

Mori dream spaces and GIT.

The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the chamber decomposition of the cone of divisors in Mori theory is naturally identified with the decomposition of the G-ample cone from geometric invariant theory. In particular every rational contraction of a Mori dream space comes from GIT, and all possible factorizations of a rational contraction can be read off from the chamber decomposition.

A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations

We briefly review the formal picture in which a Calabi-Yau $n$-fold is the complex analogue of an oriented real $n$-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of \cite{LT}, \cite{BF} in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in $\Pee^3$, and Donaldson-- and Gromov-Witten-- like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold $X$, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general $K3$ fibration $X$, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the ``Mukai-dual'' 3-fold for others. As an example the invariant is shown to distinguish Gross' diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of $X$.

Vector bundles on ruled Fano 3-folds

Let be a ruled Fano 3-fold. The goal of this paper is to compute the dimension, prove the irreducibility and smoothness and describe the structure of the moduli space M L (2;c 1,c 2) of L-stable, rank 2 vector bundles E on X with certain Chern classes and for a suitable polarization L closely related to c 2. More precisely, we will cover the study of some moduli spaces M L (2;c 1,c 2) such that the generic point is given as a non-trivial extension of line bundles. This work nicely reflects the general philosophy that moduli spaces inherits a lot of geometrical properties of the underlying variety.

On the classification of toric Fano 4-folds

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ d , the so-called Fano polyhedra, up to an isomorphism of the standard lattice % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqeeuuDJXwAKb% sr4rNCHbacfaGae8hjHi6aaWbaaSqabeaaieGacaGFKbaaaOGaeyOG% IWSae8xhHe6aaWbaaSqabeaacaGFKbaaaaaa!418C! $$\mathbb{Z}^d \subset \mathbb{R}^d$$ . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism.

Fano 4-Folds with scroll structure

Abstract.We classify smooth Fano 4-folds with second betti number b2 = 2, possesing adjunction theoretic scroll structure. These manifolds occur to be projectivisations of coherent sheaves over Fano manifolds. The paper deals mainly with projectivisations of non-locally free rank 2 Fano sheaves over Fano 3-folds with b2 = 1, using Mori theory. By the way, we classify nef and big rank 2 bundles over P2 with the first Chern class (3).

A Vector Bundle of Rank 2 on P1 ×P3

It is well known that the Horrocks–Mumford bundle [Fscr ] encodes a lot of very interesting geometric information. This is essentially the reason for the fact that much work has been done in order to find other rank-2 bundles on ℙ4. The only nonsplit vector bundles of rank 2 on ℙ4, known up to now, are twists of pullbacks of [Fscr ] by finite coverings f[ratio ]ℙ4→ℙ4. So it seems to be a natural question to consider, instead of ℙ4, other Fano 4-folds. It is the aim of this note to give an example of a rank-2 vector bundle on ℙ1×ℙ3 and to show that it also admits very interesting geometric properties.

Rigidity properties of imprimitive Fano 3-folds

Rigidity properties of imprimitive Fano 3-folds

Rank 2 stable vector bundles on Fano 3-folds of index 2

Let X be a Fano 3-fold of the first kind with index 2. In this paper, we characterize the chern classes of rank 2 stable vector bundles on X and we find a bound for the least twist of a rank 2 reflexive sheaf on X which has a global section.

On the stability of the tangent bundle of Fano manifolds

By using the classification of Fano 3-folds we prove: Let $X$ be Fano 3-fold. Assume that the tangent bundle $T_X$ of $X$ is not stable (i.e. semi-stable or unstable). Then $b_2\geq 2$ and a relative tangent sheaf $T_{X/Y}$ of a contraction $f:X\longrightarrow Y$ of an extremal face on $X$ is a destabilising subsheaf of $T_X$.

The fundamental group of the smooth part of a log Fano variety

Let X be a normal projective variety over the complex number field C. We call X a Fano variety if X is Q-Gorenstein and the anti-canonical divisor — Kx is ample. A Fano variety X is said to be a log Fano variety if X has only log terminal singularities (cf. [6]). A Fano variety X is called a canonical Fano variety if X has only canonical singularities (cf. [6]). The Cartier index c{X) is the smallest positive integer such that c{X)Kx is a Cartier divisor. The Fano index, denoted by r(X), is the largest positive rational number such that —Kχ~~Q r{X)H (Q-linear equivalence) for a Cartier divisor H. This note consists of two sections. In §1, we shall consider canonical Fano 3-folds and prove the following:

On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians

On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians

On fano 3-folds with non-rational singularities and two-dimensional base

On fano 3-folds with non-rational singularities and two-dimensional base

A Fano $3$-fold with non-rational singularities and a two-dimensional basis

In this paper, the author gives a summary of the paper [12] and pictures of theFano 3-folds which appear in it. Here a Fano 3-fold means a normal projective varietyof dimension three over C whose anticanonical sheaf is ample and invertible. During thepast fifteen years, there has been big progress in the investigation of a non-singular Fano3-fold owing to Iskovskih, Mori, Mukai and Shokurov. And it is still developing. On theother hand, in singular Fano 3-folds, progress seems to have started recently. Here westudy the structure of a Fano 3-fold with non-rational singularities.Let Σ be the locus of non-rational singular points of a Fano 3-fold