Fano Surface Research Articles

Remark on complements on surfaces

Abstract We give an explicit characterization on the singularities of exceptional pairs in any dimension. In particular, we show that any exceptional Fano surface is $\frac {1}{42}$ -lc. As corollaries, we show that any $\mathbb R$ -complementary surface X has an n-complement for some integer $n\leq 192\cdot 84^{128\cdot 42^5}\approx 10^{10^{10.5}}$ , and Tian’s alpha invariant for any surface is $\leq 3\sqrt {2}\cdot 84^{64\cdot 42^5}\approx 10^{10^{10.2}}$ . Although the latter two values are expected to be far from being optimal, they are the first explicit upper bounds of these two algebraic invariants for surfaces.

A note on Bridgeland moduli spaces and moduli spaces of sheaves on X_{14} and Y_3

We study Bridgeland moduli spaces of semistable objects of (-1)-classes and (-4)-classes in the Kuznetsov components on index one prime Fano threefold X_{4d+2} of degree 4d+2 and index two prime Fano threefold Y_d of degree d for d=3,4,5. For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of (-1)-classes on X_{4d+2} and Y_d are isomorphic. We show that moduli spaces of stable objects of (-1)-classes on X_{14} are realized by Fano surface mathcal {C}(X) of conics, moduli spaces of semistable sheaves M_X(2,1,6) and M_X(2,-1,6) and the correspondent moduli spaces on cubic threefold Y_3 are realized by moduli spaces of stable vector bundles M^b_Y(2,1,2) and M^b_Y(2,-1,2). We show that moduli spaces of semistable objects of (-4)-classes on Y_{d} are isomorphic to the moduli spaces of instanton sheaves M^{inst}_Y when dne 1,2, and show that there are open immersions of M^{inst}_Y into moduli spaces of semistable objects of (-4)-classes when d=1,2. Finally, when d=3,4,5 we show that these moduli spaces are all isomorphic to M^{ss}_X(2,0,4).

Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic threefolds

We focus on a class of Weierstraß elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically K-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform of a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We also show that the inverse Fourier-Mukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to modification in codimension 2.

Summands of theta divisors on Jacobians

We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

Deformation Classification of Real Non-singular Cubic Threefolds with a Marked Line

We prove that the space of pairs (X, l) formed by a real non-singular cubic hypersurface $$X\subset P^4$$ with a real line $$l\subset X$$ has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface $$F_{\mathbb R}(X)$$ formed by real lines on X. For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of X characterizes completely the component.

Measures of irrationality of the Fano surface of a cubic threefold

For $X$ a smooth cubic threefold, we study the Plücker embedding of the Fano surface of lines $S$ of $X$. We prove that if $X$ is general, then the minimal gonality of a covering family of curves of $S$ is four, and that this happens for a unique family of curves. The analysis also shows that there is a unique pentagonal connecting family of curves, which leads to the fact that the connecting gonality of $S$ is five, whereas the degree of irrationality, i.e., the minimal degree of a rational map from $S$ to $\mathbb {P}^2$, is six.

The motive of the Fano surface of lines

In this short note, we prove that the Chow motive of the Fano surface of lines on the smooth cubic threefold is finite-dimensional in the sense of Kimura. This gives an example of a variety not dominated by a product of curves whose Chow motive is of Abelian type.

GV-subschemes and their embeddings in principally polarized abelian varieties

We prove that any embedding of a -subscheme in a principally polarized abelian variety does not factor through any nontrivial isogeny. As an application, we present a new proof of a theorem of Clemens–Griffiths identifying the intermediate Jacobian of a smooth cubic threefold to the Albanese variety of its Fano surface of lines.

On the Tate conjecture for the Fano surfaces of cubic threefolds

A Fano surface of a smooth cubic threefold X ↪ P 4 parametrizes the lines on X . In this note, we prove that a Fano surface satisfies the Tate conjecture over a field of finite type over the prime field and characteristic not 2.

The Kähler Ricci flow on Fano manifolds (I)

We study the evolution of pluri-anticanonical line bundles K_M^{-\nu} along the Kähler Ricci flow on a Fano manifold M . Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of M . For example, the Kähler Ricci flow on M converges when M is a Fano surface satisfying c_1^2(M)=1 or c_1^2(M)=3 . Combined with the works in [CW1] and [CW2], this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups. The original proof of this conjecture is due to Gang Tian in [Tian90].

The Fano normal function

The Fano surface F of lines in the cubic threefold V is naturally embedded in the intermediate Jacobian J(V), we call “Fano cycle” the difference F−F−, this is homologous to 0 in J(V). We study the normal function on the moduli space which computes the Abel–Jacobi image of the Fano cycle. By means of the related infinitesimal invariant we can prove that the primitive part of the normal function is not of torsion. As a consequence we get that, for a general V, F−F− is not algebraically equivalent to zero in J(V) (proved also by van der Geer and Kouvidakis (2010) [15] with different methods) and, moreover, that there is no divisor in JV containing both F and F− and such that these surfaces are homologically equivalent in the divisor.Our study of the infinitesimal variation of Hodge structure for V produces intrinsically a threefold Ξ(V) in the Grassmannian of lines G in P4. We show that the infinitesimal invariant at V attached to the normal function gives a section of a natural bundle on Ξ(V) and more specifically that this section vanishes exactly on Ξ∩F, which turns out to be the curve in F parameterizing the “double lines” in the threefold. We prove that this curve reconstructs V and hence we get a Torelli-like result: the infinitesimal invariant for the Fano cycle determines V.

Fano threefolds of Genus 6

Ideas and methods of Clemens C. H., Griffiths Ph. The intermediate Jacobian of a cubic threefold are applied to a Fano threefold $X$ of genus 6 — intersection of $G(2, 5) \subset P^9$ with $P^7$ and a quadric. Main results: 1. The Fano surface $F(X)$ of $X$ is smooth and irreducible. Hodge numbers and some other invariants of $F(X)$ are calculated. 2. Tangent bundle theorem for $X$ is proved, and its geometric interpretation is given. It is shown that $F(X)$ defines $X$ uniquely. 3. The Abel-Jacobi map $\Phi : \operatorname{Alb} F(X) \to J^3(X)$ is an isogeny. 4. As a necessary step of calculation of $h^{1,0}(F(X))$ we describe a special intersection of 3 quadrics in $P^6$ (having 1 double point) whose Hesse curve is a smooth plane curve of degree 6. 5. $\operatorname{im} \Phi(F(X)) \subset J^3(X)$ is algebraically equivalent to $\frac{2\Theta^8}{8!}$ where $\Theta \subset J^3(X)$ is a Poincare divisor (a sketch of the proof).

Kähler–Einstein metrics on Fano surfaces

We give an exposition of a result of G. Tian, which says that a Fano surface admits a Kähler–Einstein metric precisely when the Lie algebra of holomorphic vector fields is reductive.

Fano surfaces with 12 or 30 elliptic curves

Curves of low genus on a surface carry important informations on that surface. We study the Fano surfaces of lines of cubic threefolds that contain 12 or 30 elliptic curves. We determine their Picard number and compute a basis of the N\'eron-Severi group of the Fano surface of the Fermat cubic threefold.

The Fano surface of the Fermat cubic threefold, the del Pezzo surface of degree $5$ and a ball quotient

In this paper we study a surface which has many intriguing and puzzling aspects: on one hand it is related to the Fano surface of lines of a cubic threefold, and on the other hand it is related to a ball quotient occurring in the realm of hypergeometric functions, as studied by Deligne and Mostow. It is moreover connected to a surface constructed by Hirzebruch in his works for constructing surfaces with Chern ratio equal to 3 by arrangements of lines on the plane. Furthermore, we obtain some results that are analogous to the results of Yamasaki-Yoshida when they computed the lattice of the Hirzebruch ball quotient surface.

The Kähler Ricci flow on Fano surfaces (I)

Suppose {(M, g(t)), 0 ≤ t < ∞} is a Kahler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kahler Ricci flow on toric Fano surface, whose initial metric has toric symmetry. In particular, such a Kahler Ricci flow must converge to a Kahler Ricci soliton metric. Therefore we give a new Ricci flow proof of the existence of Kahler Ricci soliton metrics on toric Fano surfaces.

MINIMAL CLASSES ON THE INTERMEDIATE JACOBIAN OF A GENERIC CUBIC THREEFOLD

Let X be a smooth cubic threefold. We can associate two objects to X: the intermediate Jacobian J and the Fano surface F parametrizing lines on X. By a theorem of Clemens and Griffiths, the Fano surface can be embedded in the intermediate Jacobian and the cohomology class of its image is minimal. In this paper, we show that if X is generic, the Fano surface is the only surface of minimal class in J.

Elliptic curve configurations on Fano surfaces

The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces.

The Fano surface of the Klein cubic threefold

We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order $11$. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index $2$ sub-group of the Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.

Cubic hypersurfaces and integrable systems

Together with the cubic and quartic threefolds, the cubic fivefolds are the only hypersurfaces of odd dimension bigger than one for which the intermediate Jacobian is a nonzero principally polarized abelian variety (p.p.a.v.). In this paper we show that the family of $21$-dimensional intermediate Jacobians of cubic fivefolds containing a given cubic fourfold $X$ is generically an algebraic integrable system. In the proof we apply an integrability criterion, introduced and used by Donagi and Markman to find a similar integrable system over the family of cubic threefolds in $X$. To enter in the conditions of this criterion, we write down explicitly the symplectic structure, known by Beauville and Donagi, on the family $F(X)$ of lines on the general cubic fourfold $X$, and prove that the family of planes on a cubic fivefold containing $X$ is embedded as a Lagrangian surface in $F(X)$. By a symplectic reduction we deduce that our integrable system induces on the nodal boundary another integrable system, interpreted generically as the family of $20$-dimensional intermediate Jacobians of Fano threefolds of genus four contained in $X$. Along the way we prove an Abel-Jacobi type isomorphism for the Fano surface of conics in the general Fano threefold of genus four, and compute the numerical invariants of this surface.