arXiv:2003.08795v3 [math.AG] 29 Sep 2020

ON FANO SCHEMES OF LINEAR SPACES OF GENERAL COMPLETE INTERSECTIONS

FRANCESCO BASTIANELLI, CIRO CILIBERTO, FLAMINIO FLAMINI, AND PAOLA SUPINO

Abstract. We consider the Fano scheme Fk(X) of k–dimensional linear subspaces contained in
a complete intersection X ⊂ P^{n}of multi–degree d = (d1, . . . , ds). 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 Π^{s}_{i=1}di > 2 and we find
conditions on n, d and k under which Fk(X) does not contain either rational or elliptic curves. At
the end of the paper, we study the case Π^{s}_{i=1}di= 2.

1. Introduction

*In this paper we are concerned with the Fano scheme F*_{k}(X) ⊂ G(k, n), parameterizing k–

dimensional linear subspaces contained in X ⊂ P^{n}, when X is a complete intersection of multi–

degree d = (d_{1}, . . . , d_{s}), with 1 6 s 6 n − 2. We will avoid the trivial case in which X is a linear
subspace, so that Π^{s}_{i=1} d_{i} >2.

Our inspiration has been the following result by Riedl and Yang concerning the case of hyper- surfaces:

Theorem 1.1. *(cf. [8, Thm. 3.3]) If X ⊂ P*^{n} *is a very general hypersurface of degree d such that*
n 6 ^{(d+1)(d+2)}_{6} *, then F*_{1}*(X) contains no rational curves.*

This paper is devoted to generalize Riedl–Yang’s result to complete intersections and to arbi- trary k>1:

Theorem 1.2. *Let X ⊂ P*^{n}*be a very general complete intersection of multi–degree d = (d*_{1}, . . . , d_{s}*),*
*with 1 6 s 6 n − 2 and Π*^{s}_{i=1}d_{i}>*2.*

*Let 1 6 k 6 n − s − 1 be an integer. If*

n 6 k− 1 + 1 k+ 2

Xs i=1

d_{i}+ k + 1
k+ 1

, (1.1)

*then F*k*(X) contains neither rational nor elliptic curves.*

The proof is contained in Section 3. Section 4 concerns the quadric case Π^{s}_{i=1}d_{i} = 2.

*We work over the complex field C. As customary, the term “general” is used to denote a point*
which sits outside a union of finitely many proper closed subsets of an irreducible algebraic variety
*whereas the term “very general” is used to denote a point sitting outside a countable union of*
proper closed subsets of an irreducible algebraic variety.

Key words and phrases. Complete intersections; parameter spaces; Fano schemes; rational and elliptic curves.

Mathematics Subject Classification (2010). Primary 14M10; Secondary 14C05, 14M15, 14M20.

This collaboration has benefitted of funding from the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP: E83-C18000100006).

1

2. Preliminaries

Let n > 3, 1 6 s 6 n − 2 and d = (d_{1}, . . . , ds) be an s–tuple of positive integers such
that Π^{s}_{i=1} d_{i} > 2. Let S_{d} := Ls

i=1H^{0}(P^{n},OP^{n}(d_{i})) and consider its Zariski open subset S_{d}^{∗} :=

Ls

i=1 H^{0}(P^{n},OP^{n}(d_{i})) \ {0}

. For any u := (g_{1}, . . . , g_{s}) ∈ S_{d}^{∗}, let X_{u} := V (g_{1}, . . . , g_{s}) ⊂ P^{n}
denote the closed subscheme defined by the vanishing of the polynomials g_{1}, . . . , gs. When u ∈ S_{d}^{∗}
is general, X_{u} is a smooth, irreducible variety of dimension n − s > 2, so that S_{d}^{∗} contains an open
dense subset parameterizing s–tuples u such that Xu is a smooth complete intersection.

For any integer 1 6 k 6 n − s − 1, consider the locus
W_{d,k}:=n

u∈ S_{d}^{∗}

F^{k}(X_{u}) 6= ∅o

⊆ S_{d}^{∗}
and set

t(n, k, d) := (k + 1)(n − k) − Xs

i=1

d_{i}+ k
k

.

If no confusion arises, we will denote by t the integer t(n, k, d). In this set–up, we remind the following results:

Result 2.1. *(cf., e.g., [1, 2, 5, 6, 7] and [3, Thm. 22.14, p. 294]) Let n, k, s and d = (d*_{1}, . . . , d_{s}*) be*
*integers as above.*

*(a) When Π*^{s}_{i=1}d_{i} >*2, one has the following situation.*

*(i) For t < 0, W*d,k ( S_{d}^{∗} *so, for u ∈ S*_{d}^{∗} *general, F*k(Xu*) = ∅.*

*(ii) For t > 0, W*_{d,k}= S^{∗}_{d} *and, for u ∈ S*_{d}^{∗} *general, F*_{k}(X_{u}*) is smooth with dim(F*_{k}(X_{u}*)) = t and it*
*is irreducible when t > 1.*

*(b) When Π*^{s}_{i=1}d_{i}*= 2, one has the following situation.*

*(i) For ⌊*^{n−s}_{2} *⌋ < k 6 n − s − 1, W*d,k ( S_{d}^{∗}*; more precisely, for u ∈ S*_{d}^{∗} *such that X*u *is smooth,*
*one has F*_{k}(X_{u}*) = ∅.*

*(ii) For 1 6 k 6 ⌊*^{n−s}_{2} *⌋, W*d,k = S_{d}^{∗} *and, for u ∈ S*_{d}^{∗} *such that X*u *is smooth, then F*k(Xu*) is*
*smooth, (equidimensional) with dim(F*k(Xu)) = t = (k + 1)(n − s − ^{3k}_{2}*). Moreover F*k(Xu)
*is irreducible unless dim(X*u*) = n − s is even and k =* ^{n−s}_{2} *, in which case F*k(Xu*) consists*
*of two disjoint irreducible components. In this case, each irreducible component of F*_{k}(X_{u}*) is*
*isomorphic to F*_{k−1}(X_{u}^{′}*), where X*_{u}^{′} *is a general hyperplane section of X*u*.*

Result 2.2. *(cf., e.g., [2, Rem. 3.2, (2)]) Let n, k, s and d = (d*_{1}, . . . , d_{s}*) be integers as above with*
Π^{s}_{i=1}d_{i} >*2. When u ∈ S*_{d}^{∗} *is such that X*_{u} ⊂ P^{n} *is a smooth, irreducible complete intersection of*
*dimension n − s and F*k(Xu*) is not empty, smooth and of (expected) dimension t, the canonical*
*bundle of F*_{k}(Xu*) is given by*

ω_{F}_{k}_{(X}_{u}_{)}= O_{F}_{k}_{(X}_{u}_{)} −n − 1 +
Xs
i=1

d_{i}+ k
k+ 1

!

(2.1)

*where O*_{F}_{k}_{(X}_{u}_{)}*(1) is the hyperplane line bundle of F*_{k}(Xu*) in P(*^{n+1}^{k+1})^{−}1 *via the Pl¨**ucker embedding*
*of G(k, n).*

We will also need the following:

Result 2.3. *(cf.,[8, Prop. 3.5]) Let n, m be positive integers with m 6 n. Let B ⊂ G(m, n) be*
*an irreducible subvariety of codimension at least ǫ > 1. Let C ⊂ G(m − 1, n) be a non–empty*
*subvariety satisfying the following condition: for every (m − 1)-plane c ∈ C, if b ∈ G(m, n) is such*
*that c ⊂ b, then b ∈ B. Then the codimension of C in G(m − 1, n) is at least ǫ + 1.*

3. The proof of Theorem 1.2

This section is devoted to the proof of Theorem 1.2, which uses a strategy similar to the one in [8, Thm. 3.3].

*Proof of Theorem 1.2. Let G := G(k, n) be the Grassmannian of k-planes in P*^{n} and H := Hd,n

be the irreducible component of the Hilbert scheme whose general point parameterizes a smooth,
irreducible complete intersection X ⊂ P^{n} of dimension n − s > 2 and multi–degree d (H is the
image of an open dense subset of S^{∗}_{d} via the classifying morphism). Let us consider the incidence
correspondence

Uk,n,d:= { ([Λ] , [X]) ∈ G × H| Λ ⊂ X} ⊂ G × H
with the two projections G←− U^{π}^{1} k,n,d

π2

−→ H.

Note that U_{k,n,d}is smooth and irreducible, because the map π_{1}is surjective and has smooth and
irreducible fibres which are all isomorphic via the action of the group of projective transformations.

Since Π^{s}_{i=1}d_{i} > 2, from Result 2.1–(a), if t 6 0 then F_{k}(X) is either empty or a zero–

dimensional scheme, so in particular the statement holds true. We can therefore assume t > 1.

In this case, from Result 2.1–(a.ii), the map π_{2} is dominant and the fibre over the general point
[X] ∈ H is isomorphic to F_{k}(X), hence it is smooth, irreducible of dimension t. Thus U_{k,n,d}
dominates H via π2 and

dim(U_{k,n,d}) = dim(H) + t. (3.1)

Consider now Rk,n,d :=

([Λ], [X]) ∈ U_{k,n,d}

there exists a rational curve
in F_{k}(X) containing [Λ]

⊆ Uk,n,d

and similarly

Ek,n,d :=

([Λ], [X]) ∈ U_{k,n,d}

there exists an elliptic curve
in F_{k}(X) containing [Λ]

⊆ Uk,n,d.

Notice that both R_{k,n,d} and E_{k,n,d} are (at most) countable unions of irreducible locally closed
subvarieties. Let us see this for R_{k,n,d}, the case of E_{k,n,d}being analogous. Look at the morphism π_{2} :
Uk,n,d −→ H, whose fibre over a point [X] ∈ H is isomorphic to the Fano scheme Fk(X). Consider
the locally closed subset H in the Hilbert scheme consisting of the set of points parameterizing
irreducible rational curves in U_{k,n,d}contained in fibres of π_{2}. Then H, as well as the Hilbert scheme,
is a countable union of irreducible varieties. Consider the incidence correspondence

I =

(([Λ], [X]) , [C]) ∈ U_{k,n,d}× H

([Λ], [X]) ∈ C .

Since H is a countable union of irreducible varieties, then also I is a countable union of irreducible
varieties. Finally R_{k,n,d} is the image of I via the projection on U_{k,n,d}, and therefore it is (at most)
a countable union of irreducible varieties.

We point out that proving the assertion for very general complete intersections X ⊂ P^{n} of
multi–degree d = (d_{1}, . . . , d_{s}) is equivalent to proving it for very general complete intersections
Y ⊂ P^{M} of multi–degree

d^{n}= d^{n}_{M} := (1, . . . , 1

| {z }

M −n

, d_{1}, . . . , d_{s})

for some M > n. In particular, condition (1.1) gives equivalent inequalities and t := t(n, k, d) =
t(M, k, d^{n}_{M}). Moreover, in the light of this fact, we can assume d1, . . . , d_{s} >2.

We claim that, under our assumptions, there exists M > n such that both R_{k,M,d}^{n} and E_{k,M,d}^{n}
have codimension at least t + 1 in U_{k,M,d}^{n}, which proves the statement. Indeed, consider the case
of Rk,M,d^{n} (the same reasoning applies to the case with Ek,M,d^{n}); if codimU_{k,M,dn}(Rk,M,d^{n}) > t + 1
then, by formula (3.1), we have that dim(R_{k,M,d}^{n}) 6 dim(H_{d}^{n}_{,M}) − 1. Thus dim(π_{2} Rk,M,d^{n}

) 6
dim(Hd^{n},M) − 1, proving the assertion for very general complete intersections Y ⊂ P^{M} of multi–

degree d^{n}, and hence for very general complete intersections X ⊂ P^{n}of multi–degree d.

We are therefore left to showing that there exists M > n such that both R_{k,M,d}^{n} and E_{k,M,d}^{n}
have codimension at least t + 1 in Uk,M,d^{n}. In what follows, we will focus on Rk,M,d^{n} (the same
arguments work for the case E_{k,M,d}^{n}).

If R_{k,n,d} = ∅, we are done. So assume R_{k,n,d} 6= ∅ and take ([Λ0], [X_{0}]) ∈ R_{k,n,d} a very
general point in an irreducible component of Rk,n,d. We note that for any M > n, we can embed
X_{0} ⊂ P^{n} into a n-plane of P^{M}, and hence we can identify X_{0} to a complete intersection in P^{M} of
multi–degree d^{n}, so that ([Λ0], [X0]) ∈ Rk,M,d^{n}. Therefore it suffices to find some M > n and an
irreducible subvariety F ⊂ U_{k,M,d}^{n} such that

([Λ_{0}], [X_{0}]) ∈ F and codimF(R_{k,M,d}^{n}∩ F) > t + 1.

To do so, we start with the following remark: if X ⊂ P^{n} is a very general complete intersection of
multi–degree d, it follows from Results 2.1–(a.ii) and 2.2 and formula (2.1) that ω_{F}_{k}_{(X)} is ample if
and only if

n 6 Xs i=1

d_{i}+ k
k+ 1

− 2.

We set

m= m(d, k) :=

Xs i=1

d_{i}+ k
k+ 1

− 2
and, in view of the obvious equality ^{d}^{i}^{+k+1}_{k+1}

= ^{d}_{k+1}^{i}^{+k}

+ ^{d}^{i}_{k}^{+k}

, we notice that the hypothesis (1.1) reads m − n > t. Hence we have m > n.

Let Y^{′} ⊂ P^{m} be a very general complete intersection of multi–degree d and let Λ^{′} ⊂ Y^{′} be
a very general k-plane of Y^{′}, i.e., [Λ^{′}] corresponds to a very general point in F_{k}(Y^{′}). By Result
2.1 and the fact that m > n, we have that F_{k}(Y^{′}) is smooth, irreducible with dim(F_{k}(Y^{′})) =
(k + 1)(m − k) −Ps

i=1 di+k

k

> t >1. Notice that there are no rational curves in Fk(Y^{′}) through
[Λ^{′}] since F_{k}(Y^{′}) is smooth and of general type, because ω_{F}_{k}_{(Y}′) = O_{F}_{k}_{(Y}′)(1) by the choice of m,
and [Λ^{′}] is very general on F_{k}(Y^{′}). In other words ([Λ^{′}], [Y^{′}]) ∈ U_{k,m,d}\ Rk,m,d.

Take now an integer M ≫ m > n and let Y^{′′} ⊂ P^{M} be a smooth, complete intersection of
multi–degree d, containing a k-plane Λ^{′′}, such that X_{0} is a n-plane section of Y^{′′}, Y^{′} is a m-plane
section of Y^{′′}, and Λ^{′′}= Λ0 = Λ^{′}.

For any integer r > n, let Z_{r} ⊂ G(r, M ) be the variety of r-planes in P^{M} containing Λ_{0} and
let Z_{r}^{′} ⊆ Zr be the subset of those r-planes Λ ∈ Z_{r} such that the Fano scheme F_{k}(Y^{′′}∩ Λ) of the
complete intersection Y^{′′}∩ Λ ⊂ P^{M} contains a rational curve through the point [Λ0] ∈ Fk(Y^{′′}∩ Λ).

Note that Z_{r} is isomorphic to G(r − k − 1, M − k − 1).

For any integer r > n, we define the morphism φr: Zr −→ Uk,M,d^{r} sending the r-plane Λ to
([Λ_{0}], [Λ ∩ Y^{′′}]) ∈ U_{k,M,d}^{r}. It is clear that φr maps Zr isomorphically onto its image Fr := φr(Zr),
where the inverse map sends a pair ([Λ_{0}], [Y ]) ∈ F_{r} to the point [Λ] ∈ Z_{r} corresponding to the
linear span Λ = hY i (recall that we assumed d1, . . . , d_{s} > 2). Then the image of Z_{r}^{′} in Uk,M,d^{r} is
Fr∩ Rk,M,d^{r}. We will set F := F_{n}.

By construction of the pair ([Λ_{0}], [Y^{′}]) ∈ U_{k,M,d}^{m} \ Rk,M,d^{m}, we have that codimF_{m}(F_{m} ∩
R_{k,M,d}^{m}) = ǫ > 1. Now we apply Result 2.3, to Fm ∼= G(r−k−1, M −k−1), B = Fm∩R_{k,M,d}^{m}and
C = F_{m−1}∩ R_{k,M,d}^{m−1}, because clearly B and C verify the condition stated there. We deduce that
codimF_{m−1}(F_{m−1}∩R_{k,M,d}m−1) > ǫ+1. Iterating this argument we see that codimF(F ∩R_{k,M,d}^{n}) >

m− n + 1. Now, as we already noticed, m − n + 1 > t + 1 is equivalent to the condition (1.1),
hence we have codimF(F ∩ R_{k,M,d}^{n}) > t + 1, as wanted. This completes the proof.
*Remark 3.1. Let X ⊂ P*^{n} be a general complete intersection of multi–degree d = (d_{1}, . . . , d_{s}). If
Ps

i=1 di+k

k+1

6n,then, by (2.1), Fk*(X) is a smooth Fano variety and therefore by [4] it is rationally*
*connected, i.e., there is a rational curve passing through two general points of it.*

4. The quadric case Here we prove the following result:

Theorem 4.1. *Let X ⊂ P*^{n} *be a smooth complete intersection of multi–degree d = (d*_{1}, . . . , d_{s}*),*
*with 1 6 s 6 n − 2 and Π*^{s}_{i=1}d_{i}*= 2. Let k be an integer such that 1 6 k 6 n − s − 1. Then:*

*(i) for ⌊*^{n−s}_{2} *⌋ < k 6 n − s − 1, F*k*(X) is empty, whereas*

*(ii) for 1 6 k 6 ⌊*^{n−s}_{2} *⌋, the single component or both components of F*k*(X) (see Result 2.1–(b.ii))*
*are rationally connected.*

*Proof. Since Π*^{s}_{i=1}di = 2, we have that X ⊂ P^{n−s+1}is a smooth quadric hypersurface. From Result
2.1–(b), if ⌊^{n−s}_{2} ⌋ < k 6 n − s − 1, Fk(X) is empty.

Next we assume 1 6 k 6 ⌊^{n−s}_{2} ⌋. From Result 2.2 and formula (2.1), we have
ω_{F}_{k}_{(X)} = O_{F}_{k}_{(X)}(−n + s + k) .

Since k 6 ^{n−s}_{2} , then −n + s + k 6 −1 therefore the single component or both components of F_{k}(X)
(see Result 2.1–(b.ii)) are smooth Fano varieties, hence they are rationally connected by [4].
Acknowledgment. The authors would like to thank Lawrence Ein for having pointed out the
paper [8].

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Francesco Bastianelli, Dipartimento di Matematica, Universit`a degli Studi di Bari “Aldo Moro”, Via Edoardo Orabona 4, 70125 Bari – Italy

E-mail address: francesco.bastianelli@uniba.it

Ciro Ciliberto, Dipartimento di Matematica, Universit`a degli Studi di Roma “Tor Vergata”, Viale della Ricerca Scientifica 1, 00133 Roma – Italy

E-mail address: cilibert@mat.uniroma2.it

Flaminio Flamini, Dipartimento di Matematica, Universit`a degli Studi di Roma “Tor Vergata”, Viale della Ricerca Scientifica 1, 00133 Roma – Italy

E-mail address: flamini@mat.uniroma2.it

Paola Supino, Dipartimento di Matematica e Fisica, Universit`a degli Studi “Roma Tre”, Largo S. L. Murialdo 1, 00146 Roma – Italy

E-mail address: supino@mat.uniroma3.it