Fano Scheme Research Articles

On some invariants of cubic fourfolds

For a general cubic fourfold Xsubset mathbb {P}^5 with Fano variety F, we compute the Hodge numbers of the locus Ssubset F of lines of second type and the class of the locus Vsubset F of triple lines, using the description of the latter in terms of flag varieties. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any smooth cubic hypersurface.

Geometry of Lines on a Cubic Four-Fold

Abstract For a general cubic four-fold $X\subset {\mathbb {P}}^5$ with Fano scheme of lines $F$, we prove a number of properties of the universal family of lines $I\to F$ and various subloci. We first describe the moduli and ramification theory of the genus four fibration $p:I\to X$ and explore its relation to a birational model of $F$ in $I$. The main part of the paper is devoted to describing the locus $V\subset F$ of triple lines, i.e., the fixed locus of the Voisin map $\phi :F\dashrightarrow F$, in particular proving it is an irreducible projective singular surface of class $21\textrm {c}_2({\mathcal {U}}_F)$ and detailing its intersection with the locus $S$ of second type lines. A consequence of the analysis of the singularities of $V$ is a geometric proof of the fact that if $X$ is very general, then the number of singular (necessarily 1-nodal) rational curves in $F$ of primitive class is 3780.

38406501359372282063949 and All That: Monodromy of Fano Problems

Abstract A Fano problem is an enumerative problem of counting $r$-dimensional linear subspaces on a complete intersection in ${\mathbb{P}}^n$ over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of finite Fano schemes $F_{r}(X)$ as the complete intersection $X$ varies. We prove that the monodromy group is either symmetric or alternating in most cases. In the exceptional cases, the monodromy group is one of the Weyl groups $W(E_6)$ or $W(D_k)$.

On Fano schemes of linear spaces of general complete intersections

We consider the Fano scheme $$F_k(X)$$ of k-dimensional linear subspaces contained in a complete intersection $$X \subset {\mathbb {P}}^n$$ of multi-degree $${\underline{d}} = (d_1, \ldots , d_s)$$ . Our main result is an extension of a result of Riedl and Yang concerning Fano schemes of lines on very general hypersurfaces: we consider the case when X is a very general complete intersection and $$\Pi _{i=1}^s d_i > 2$$ and we find conditions on n, $${\underline{d}}$$ , and k under which $$F_k(X)$$ does not contain either rational or elliptic curves. At the end of the paper, we study the case $$\Pi _{i=1}^s d_i = 2$$ .

Fano schemes for generic sums of products of linear forms

We study the Fano scheme of [Formula: see text]-planes contained in the hypersurface cut out by a generic sum of products of linear forms. In particular, we show that under certain hypotheses, linear subspaces of sufficiently high dimension must be contained in a coordinate hyperplane. We use our results on these Fano schemes to obtain a lower bound for the product rank of a linear form. This provides a new lower bound for the product ranks of the [Formula: see text] Pfaffian and [Formula: see text] permanent, as well as giving a new proof that the product and tensor ranks of the [Formula: see text] determinant equal five. Based on our results, we formulate several conjectures.

On complete intersections containing a linear subspace

Consider the Fano scheme $$F_k(Y)$$ parameterizing k-dimensional linear subspaces contained in a complete intersection $$Y \subset \mathbb {P}^m$$ of multi-degree $$\underline{d} = (d_1, \ldots , d_s)$$. It is known that, if $$t := \sum _{i=1}^s \left( {\begin{array}{c}d_i +k\\ k\end{array}}\right) -(k+1) (m-k)\leqslant 0$$ and $$\prod _{i=1}^sd_i >2$$, for Y a general complete intersection as above, then $$F_k(Y)$$ has dimension $$-t$$. In this paper we consider the case $$t> 0$$. Then the locus $$W_{\underline{d},k}$$ of all complete intersections as above containing a k-dimensional linear subspace is irreducible and turns out to have codimension t in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general $$[Y]\in W_{\underline{d},k}$$ the scheme $$F_k(Y)$$ is zero-dimensional of length one. This implies that $$W_{\underline{d},k}$$ is rational.

Maximal linear spaces contained in the base loci of pencils of quadrics

The geometry of the Fano scheme of maximal linear spaces contained in the base locus of a pencil of quadrics has been studied by algebraic geometers when the base field is algebraically closed. In this paper, we work over an arbitrary base field of characteristic not equal to 2 and show how these Fano schemes are related to the Jacobians of hyperelliptic curves. In particular, if $B$ is the base locus of a generic pencil of quadrics in $\bbp^{2n+1}$, and $F$ is the Fano variety of $n - 1$ planes contained in $B$, then $F$ is a component of a disconnected commutative algebraic group $G = \picz(C) \dcup F \dcup \pico(C) \dcup F'$, where $C$ is the hyperelliptic curve defined by the discriminant form of the pencil. In the second half of this paper, we study regular pencils of quadrics, where the hyperelliptic curve defined by the discriminant is singular.

On Fano Schemes of Toric Varieties

Given a finite set of lattice points $\mathcal{A}$, we consider the associated homogeneous binomial ideal $I_\mathcal{A}$ and projective toric variety $X_\mathcal{A}$. We give a concise combinatorial description of all linear subspaces contained in the variety $X_\mathcal{A}$, or, equivalently, all solutions in linear forms to the system of binomial equations determined by $I_\mathcal{A}$. More precisely, we study the Fano scheme $\mathbf{F}_k(X_\mathcal{A})$ whose closed points correspond to $k$-dimensional linear spaces contained in $X_\mathcal{A}$. We show that the irreducible components of $\mathbf{F}_k(X_\mathcal{A})$ are in bijection to maximal Cayley structures for $\mathcal{A}$ of length at least $k$. We explicitly describe these irreducible components and their intersection behavior, characterize when $\mathbf{F}_k(X_\mathcal{A})$ is connected, and prove that if $X_\mathcal{A}$ is smooth in dimension $k$, then every component of $\mathbf{F}_k(X_\mathcal{A})$ is smooth in its reduced structure. Furthermore, in the special case $k=\dim X_\mathcal{A}-1$, we describe the nonreduced structure of $\mathbf{F}_k(X_\mathcal{A})$.

Fano schemes of lines on toric surfaces

We completely describe the Fano scheme of lines for a projective toric surface in terms of the geometry of the corresponding lattice polygon.

On the Space of Conics on Complete Intersections

We get sharp degree bound for generic smoothness and connectedness of the space of lines and conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on hypersurfaces. As a consequence, we prove that for a Fano complete intersection $$X$$ with index $$\ge 2$$ , the $$1$$ -Griffiths group generated by algebraic $$1$$ -cycles homologous to $$0$$ modulo algebraic equivalence is trivial, which is a conjecture for general rationally connected varieties.

Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category

Given a generic family $$Q$$ of 2-dimensional quadrics over a smooth 3-dimensional base $$Y$$ we consider the relative Fano scheme $$M$$ of lines of it. The scheme $$M$$ has a structure of a generically conic bundle $$M \rightarrow X$$ over a double covering $$X \rightarrow Y$$ ramified in the degeneration locus of $$Q \rightarrow Y$$ . The double covering $$X \rightarrow Y$$ is singular in a finite number of points (corresponding to the points $$y \in Y$$ such that the quadric $$Q_y$$ degenerates to a union of two planes), the fibers of $$M$$ over such points are unions of two planes intersecting in a point. The main result of the paper is a construction of a semiorthogonal decomposition for the derived category of coherent sheaves on $$M$$ . This decomposition has three components, the first is the derived category of a small resolution $$X^+$$ of singularities of the double covering $$X \rightarrow Y$$ , the second is a twisted resolution of singularities of $$X$$ (given by the sheaf of even parts of Clifford algebras on $$Y$$ ), and the third is generated by a completely orthogonal exceptional collection.

On symplectic automorphisms of hyper-Kähler fourfolds of K3[2] type

The present paper proves that finite symplectic groups of automorphisms of hyperk\"ahler fourfolds deformation equivalent to the Hilbert scheme of two points on a $K3$ surface are contained in the simple group $Co_1$. Then we give an example of a symplectic automorphism of order 11 on the Fano scheme of lines of a cubic fourfold.

On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines

Conjecture 1.1. Let Xn−1 ⊂ P n be a hypersurface of degree d ≥ n and let F(X) ⊂ G(2, n+ 1) denote the Fano scheme of lines on X. Let B ⊂ F(X) be an irreducible component of dimension at least n − 2. Let IB := {(x,E) | x ∈ X, E ∈ B, x ∈ PE}, and let π and ρ denote (respectively) the projections to X and B. Let XB = π(IB) ⊆ X and let Cx = πρ−1ρπ−1(x). Then, for all x ∈XB , Cx ∩Xsing = ∅. If we take hyperplane sections in the case d = n, then Conjecture 1.1 would imply the following, which was conjectured independently by Debarre and de Jong.

ABEL–JACOBI MAPS FOR HYPERSURFACES AND NONCOMMUTATIVE CALABI–YAU'S

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed (2n - 4)-form on the Fano scheme of lines on a (2n - 2)-dimensional hypersurface Yn of degree n. We provide several definitions of this form — via the Abel–Jacobi map, via Hochschild homology, and via the linkage class — and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Yn we show that the Fano scheme is birational to a certain moduli space of sheaves of a (2n - 4)-dimensional Calabi–Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non-Pfaffian hypersurface but the dual Calabi–Yau becomes noncommutative.

Symplectic structures on moduli spaces of sheaves via the Atiyah class

It is proven that the composition of the Yoneda coupling with the semiregularity map is a closed 2-form on moduli spaces of sheaves. Two examples are given when this 2-form is symplectic. Both of them are moduli spaces of torsion sheaves on the cubic 4-fold Y . The first example is the Fano scheme of lines in Y . Beauville and Donagi showed that it is symplectic but did not construct an explicit symplectic form on it. We prove that our construction provides a symplectic form. The other example is the moduli space of torsion sheaves which are supported on the hyperplane sections H ∩ Y of Y and are cokernels of the Pfaffian representations of H ∩ Y .

On threefolds covered by lines

A classification theorem is given of projective threefolds that are covered by the lines of a two-dimensional family, but not by a higher dimensional family. Precisely, ifX is such a threefold, let Σ denote the Fano scheme of lines onX and μ the number of lines contained inX and passing through a general point ofX. Assume that Σ is generically reduced. Then μ ≤ 6. Moreover,X is birationally a scroll over a surface (μ = 1), orX is a quadric bundle, orX belongs to a finite list of threefolds of degree at most 6. The smooth varieties of the third type are precisely the Fano threefolds with −K X = 2H X .