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 ⊂ Pnof 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 Πsi=1di > 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 Πsi=1di= 2.
1. Introduction
In this paper we are concerned with the Fano scheme Fk(X) ⊂ G(k, n), parameterizing k–
dimensional linear subspaces contained in X ⊂ Pn, when X is a complete intersection of multi–
degree d = (d1, . . . , ds), with 1 6 s 6 n − 2. We will avoid the trivial case in which X is a linear subspace, so that Πsi=1 di >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 ⊂ Pn is a very general hypersurface of degree d such that n 6 (d+1)(d+2)6 , then F1(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 ⊂ Pnbe a very general complete intersection of multi–degree d = (d1, . . . , ds), with 1 6 s 6 n − 2 and Πsi=1di>2.
Let 1 6 k 6 n − s − 1 be an integer. If
n 6 k− 1 + 1 k+ 2
Xs i=1
di+ k + 1 k+ 1
, (1.1)
then Fk(X) contains neither rational nor elliptic curves.
The proof is contained in Section 3. Section 4 concerns the quadric case Πsi=1di = 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 = (d1, . . . , ds) be an s–tuple of positive integers such that Πsi=1 di > 2. Let Sd := Ls
i=1H0(Pn,OPn(di)) and consider its Zariski open subset Sd∗ :=
Ls
i=1 H0(Pn,OPn(di)) \ {0}
. For any u := (g1, . . . , gs) ∈ Sd∗, let Xu := V (g1, . . . , gs) ⊂ Pn denote the closed subscheme defined by the vanishing of the polynomials g1, . . . , gs. When u ∈ Sd∗ is general, Xu is a smooth, irreducible variety of dimension n − s > 2, so that Sd∗ 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 Wd,k:=n
u∈ Sd∗
Fk(Xu) 6= ∅o
⊆ Sd∗ and set
t(n, k, d) := (k + 1)(n − k) − Xs
i=1
di+ 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 = (d1, . . . , ds) be integers as above.
(a) When Πsi=1di >2, one has the following situation.
(i) For t < 0, Wd,k ( Sd∗ so, for u ∈ Sd∗ general, Fk(Xu) = ∅.
(ii) For t > 0, Wd,k= S∗d and, for u ∈ Sd∗ general, Fk(Xu) is smooth with dim(Fk(Xu)) = t and it is irreducible when t > 1.
(b) When Πsi=1di= 2, one has the following situation.
(i) For ⌊n−s2 ⌋ < k 6 n − s − 1, Wd,k ( Sd∗; more precisely, for u ∈ Sd∗ such that Xu is smooth, one has Fk(Xu) = ∅.
(ii) For 1 6 k 6 ⌊n−s2 ⌋, Wd,k = Sd∗ and, for u ∈ Sd∗ such that Xu is smooth, then Fk(Xu) is smooth, (equidimensional) with dim(Fk(Xu)) = t = (k + 1)(n − s − 3k2). Moreover Fk(Xu) is irreducible unless dim(Xu) = n − s is even and k = n−s2 , in which case Fk(Xu) consists of two disjoint irreducible components. In this case, each irreducible component of Fk(Xu) is isomorphic to Fk−1(Xu′), where Xu′ is a general hyperplane section of Xu.
Result 2.2. (cf., e.g., [2, Rem. 3.2, (2)]) Let n, k, s and d = (d1, . . . , ds) be integers as above with Πsi=1di >2. When u ∈ Sd∗ is such that Xu ⊂ Pn is a smooth, irreducible complete intersection of dimension n − s and Fk(Xu) is not empty, smooth and of (expected) dimension t, the canonical bundle of Fk(Xu) is given by
ωFk(Xu)= OFk(Xu) −n − 1 + Xs i=1
di+ k k+ 1
!
(2.1)
where OFk(Xu)(1) is the hyperplane line bundle of Fk(Xu) in P(n+1k+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 Pn and H := Hd,n
be the irreducible component of the Hilbert scheme whose general point parameterizes a smooth, irreducible complete intersection X ⊂ Pn 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←− Uπ1 k,n,d
π2
−→ H.
Note that Uk,n,dis smooth and irreducible, because the map π1is surjective and has smooth and irreducible fibres which are all isomorphic via the action of the group of projective transformations.
Since Πsi=1di > 2, from Result 2.1–(a), if t 6 0 then Fk(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 π2 is dominant and the fibre over the general point [X] ∈ H is isomorphic to Fk(X), hence it is smooth, irreducible of dimension t. Thus Uk,n,d dominates H via π2 and
dim(Uk,n,d) = dim(H) + t. (3.1)
Consider now Rk,n,d :=
([Λ], [X]) ∈ Uk,n,d
there exists a rational curve in Fk(X) containing [Λ]
⊆ Uk,n,d
and similarly
Ek,n,d :=
([Λ], [X]) ∈ Uk,n,d
there exists an elliptic curve in Fk(X) containing [Λ]
⊆ Uk,n,d.
Notice that both Rk,n,d and Ek,n,d are (at most) countable unions of irreducible locally closed subvarieties. Let us see this for Rk,n,d, the case of Ek,n,dbeing analogous. Look at the morphism π2 : 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 Uk,n,dcontained in fibres of π2. Then H, as well as the Hilbert scheme, is a countable union of irreducible varieties. Consider the incidence correspondence
I =
(([Λ], [X]) , [C]) ∈ Uk,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 Rk,n,d is the image of I via the projection on Uk,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 ⊂ Pn of multi–degree d = (d1, . . . , ds) is equivalent to proving it for very general complete intersections Y ⊂ PM of multi–degree
dn= dnM := (1, . . . , 1
| {z }
M −n
, d1, . . . , ds)
for some M > n. In particular, condition (1.1) gives equivalent inequalities and t := t(n, k, d) = t(M, k, dnM). Moreover, in the light of this fact, we can assume d1, . . . , ds >2.
We claim that, under our assumptions, there exists M > n such that both Rk,M,dn and Ek,M,dn have codimension at least t + 1 in Uk,M,dn, which proves the statement. Indeed, consider the case of Rk,M,dn (the same reasoning applies to the case with Ek,M,dn); if codimUk,M,dn(Rk,M,dn) > t + 1 then, by formula (3.1), we have that dim(Rk,M,dn) 6 dim(Hdn,M) − 1. Thus dim(π2 Rk,M,dn
) 6 dim(Hdn,M) − 1, proving the assertion for very general complete intersections Y ⊂ PM of multi–
degree dn, and hence for very general complete intersections X ⊂ Pnof multi–degree d.
We are therefore left to showing that there exists M > n such that both Rk,M,dn and Ek,M,dn have codimension at least t + 1 in Uk,M,dn. In what follows, we will focus on Rk,M,dn (the same arguments work for the case Ek,M,dn).
If Rk,n,d = ∅, we are done. So assume Rk,n,d 6= ∅ and take ([Λ0], [X0]) ∈ Rk,n,d a very general point in an irreducible component of Rk,n,d. We note that for any M > n, we can embed X0 ⊂ Pn into a n-plane of PM, and hence we can identify X0 to a complete intersection in PM of multi–degree dn, so that ([Λ0], [X0]) ∈ Rk,M,dn. Therefore it suffices to find some M > n and an irreducible subvariety F ⊂ Uk,M,dn such that
([Λ0], [X0]) ∈ F and codimF(Rk,M,dn∩ F) > t + 1.
To do so, we start with the following remark: if X ⊂ Pn 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 ωFk(X) is ample if and only if
n 6 Xs i=1
di+ k k+ 1
− 2.
We set
m= m(d, k) :=
Xs i=1
di+ k k+ 1
− 2 and, in view of the obvious equality di+k+1k+1
= dk+1i+k
+ dik+k
, we notice that the hypothesis (1.1) reads m − n > t. Hence we have m > n.
Let Y′ ⊂ Pm 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 Fk(Y′). By Result 2.1 and the fact that m > n, we have that Fk(Y′) is smooth, irreducible with dim(Fk(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 Fk(Y′) is smooth and of general type, because ωFk(Y′) = OFk(Y′)(1) by the choice of m, and [Λ′] is very general on Fk(Y′). In other words ([Λ′], [Y′]) ∈ Uk,m,d\ Rk,m,d.
Take now an integer M ≫ m > n and let Y′′ ⊂ PM be a smooth, complete intersection of multi–degree d, containing a k-plane Λ′′, such that X0 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 Zr ⊂ G(r, M ) be the variety of r-planes in PM containing Λ0 and let Zr′ ⊆ Zr be the subset of those r-planes Λ ∈ Zr such that the Fano scheme Fk(Y′′∩ Λ) of the complete intersection Y′′∩ Λ ⊂ PM contains a rational curve through the point [Λ0] ∈ Fk(Y′′∩ Λ).
Note that Zr is isomorphic to G(r − k − 1, M − k − 1).
For any integer r > n, we define the morphism φr: Zr −→ Uk,M,dr sending the r-plane Λ to ([Λ0], [Λ ∩ Y′′]) ∈ Uk,M,dr. It is clear that φr maps Zr isomorphically onto its image Fr := φr(Zr), where the inverse map sends a pair ([Λ0], [Y ]) ∈ Fr to the point [Λ] ∈ Zr corresponding to the linear span Λ = hY i (recall that we assumed d1, . . . , ds > 2). Then the image of Zr′ in Uk,M,dr is Fr∩ Rk,M,dr. We will set F := Fn.
By construction of the pair ([Λ0], [Y′]) ∈ Uk,M,dm \ Rk,M,dm, we have that codimFm(Fm ∩ Rk,M,dm) = ǫ > 1. Now we apply Result 2.3, to Fm ∼= G(r−k−1, M −k−1), B = Fm∩Rk,M,dmand C = Fm−1∩ Rk,M,dm−1, because clearly B and C verify the condition stated there. We deduce that codimFm−1(Fm−1∩Rk,M,dm−1) > ǫ+1. Iterating this argument we see that codimF(F ∩Rk,M,dn) >
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 ∩ Rk,M,dn) > t + 1, as wanted. This completes the proof. Remark 3.1. Let X ⊂ Pn be a general complete intersection of multi–degree d = (d1, . . . , ds). 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 ⊂ Pn be a smooth complete intersection of multi–degree d = (d1, . . . , ds), with 1 6 s 6 n − 2 and Πsi=1di= 2. Let k be an integer such that 1 6 k 6 n − s − 1. Then:
(i) for ⌊n−s2 ⌋ < k 6 n − s − 1, Fk(X) is empty, whereas
(ii) for 1 6 k 6 ⌊n−s2 ⌋, the single component or both components of Fk(X) (see Result 2.1–(b.ii)) are rationally connected.
Proof. Since Πsi=1di = 2, we have that X ⊂ Pn−s+1is a smooth quadric hypersurface. From Result 2.1–(b), if ⌊n−s2 ⌋ < k 6 n − s − 1, Fk(X) is empty.
Next we assume 1 6 k 6 ⌊n−s2 ⌋. From Result 2.2 and formula (2.1), we have ωFk(X) = OFk(X)(−n + s + k) .
Since k 6 n−s2 , then −n + s + k 6 −1 therefore the single component or both components of Fk(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
