Effective Estimates for Unirationality

Lorenzo Ramero

In this work we refine some estimates that where set up by Morin and Predonzan in [1] and [2]. While the language has been updated, most of the ideas go back to these authors and we have essentially added some observations in the main stream of the proofs.

1. Introduction.

Througout the paper we tacitly assume k to be an algebraically closed field and that all the varieties be defined over k. We deal with the problem of unirationality of varieties V that are complete intersection of a fixed number of hypersuperfaces V_{1}, ..., V_{m} of a projective space P_{k}^{r}, of respective degrees d_{1}, ..., d_{m}. According to [2], there exists a r_{0} = r_{0}(d_{1}, ..., d_{m}) such that for r > r_{0} the generic complete intersection V ⊆ P_{k}^{r} is unirational over the field k. Actually, one could say a little more : assuming only that k is an arbitrary field and going through the proofs with the due care one finds an explicit finite extension of k over which V is unirational. But we won’t develop these arguments, that however are quite trivial, and can be found in the related paper [3] by Ciliberto or in [4] by Murre. Our main concern will be to find an explicit formula for r_{0}. It is an open problem the determination of the best value for r_{0}, that is the lowest one, but we will determine one that, in the general case, is at least lower than the previously known (to the author!) values. We point out that in some special cases much sharper estimates are known. In particular one can see the case of hypersurfaces that is dealt with in [3].

2. Some notation and conventions.

In what follows, the symbol Gr(q, r) will denote the Grassman variety of qdimensional linear subspaces in P_{k}^{r}. Gr(q, r) is the basis of the family Gr(q, r) ⊂ Gr(q, r) × P_{k}^{r} of all the qdimensional linear subspaces of P_{k}^{r}. This is the closed set of Gr(q, r) × P_{k}^{r} with the property that for each ℓ ∈ Gr(q, r), the intersection Gr(q, r) ∩ ℓ × P_{k}^{r} is the linear space that has ℓ as Grassman coordinate.

We introduce a multi-index notation : let I = (i_{0}, .., i_{n}), where i_{0}, ..., i_{n} are nonnegative integers, and denote by x^{I} the monomial x_{0}^{i_{o}} · ... · x_{n}^{i_{n}}. Further, we write |I| = i_{0} + ... + i_{n}. Then the generic hypersurface of degree d in a projective space P_{k}^{n} has equation

∑_{|I|=d} a_{I}x^{I} = 0.

The “parameters” a_{I} satisfy the obvious restriction of not being all equal to 0, and multiplying the a_{I}’s by the same costant gives a new equation for the same variety. That is, the parameters a_{I}’s are naturally defined as homogeneous coordinates in a projective space P_{k}^{N} of dimension N = N(n, d) = \binom{n+d}{d} − 1. So it is naturally defined a family of varieties Hyp_{n,d} by the equation

Hyp_{n,d} = {((x_{0}, ..., x_{n}), (a_{0}, ..., a_{N})) ∈ P_{k}^{n} × P_{k}^{N}| ∑_{|I|=d} a_{I}x^{I} = 0} .

The same way, we can build a family parametrizing the varieties that result as complete intersection of m hypersurfaces V_{1}, ..., V_{m} of P_{k}^{n}, of respective degrees d_{1}, ..., d_{m}. We will denote this family by Int(n, d_{1}, ..., d_{m}). A generic complete intersection V = V_{1} ∩ ... ∩ V_{m} will be denoted V^{(r,d_{1},...,d_{m})}. The basis of the family will be the space of the parameters a_{I,j}, so that for each fixed j_{0} ≤ m, the a_{I,j_{0}} are coefficients in an equation for V_{j_{0}}. In other words, the basis of the family Int is the projective variety

Int(n, d_{1}, ..., d_{m}) = P_{k}^{N(n,d_{1})} × ... × P_{k}^{N(n,d_{m})}.

Every fiber of the family is a V^{(r,d_{1},...,d_{m})} with a fixed choice of hypersurfaces V_{1}, ..., V_{m} such that V is the complete intersection of the V_{i}’s. (Just pick the corresponding point in Int!).

More generally, we are going to define a family of complete intersections with frame as any family F → B that is isomorphic to a pull-back of the family Int :

\begin{aligned}F & → & Int \\ ↓ & & ↓ \\ B & → & Int \end{aligned}

In this way, Int becomes (tautologically), a universal family for the families of complete intersections with frame.

3. A Theorem of Predonzan

We state now a theorem that will be used in the proof of the main result. The theorem was originally proved by Predonzan in [5], but one can see the paper [4] of Murre for a modern account at least in the case of hypersurfaces (There is only a little extra work to settle the general case, and this can be found in [6]).

Let r be a positive integer and d_{1}, ..., d_{m} positive integers such that d_{1} ≤ ... ≤ d_{m}. Set D = (d_{1}, ..., d_{m}). We set the Reye congruence :

Rey(r, D, q) = {(V, ℓ) ∈ Int(r, D) × Gr(q, r)|ℓ ⊂ V }.

The Reye family Rey(r, D, q) is nothing but the fiber product

Rey → Int
↓  ↓
Rey → Int

Sometimes we will omit the parameters, when there is no danger of confusion. Theorem 1. Suppose that D ≠ (1, 1, ..., 1, 2). Then the natural projection

\begin{matrix}Rey(r, D, q) \\ ↓ \\ Int(r, D) \end{matrix}

is surjective if and only if the following inequality holds:

(r − q)(q + 1) − ∑_{j=1}^{m} \binom{d_{j} + q}{q} ≥ 0.  (∗)

We denote by e(D, q) the smallest integer r that verifies (∗).

4. An improvement upon a theorem of Morin.

Suppose, as above, 1 ≤ d_{1} ≤ ... ≤ d_{m} and set D = (d_{1}, ..., d_{m}). We shall denote by D − 1 the multi-index (d_{1} − 1, ..., d_{m} − 1). We introduce the functions r(D) and w(D) by the recursive equations :

\begin{matrix}− & w(1) = 0;  r(1) = 1 \\ − & w(2) = 0;  r(2) = 2 \\ − & w(1, d_{2}, ..., d_{m}) = w(d_{2}, ..., d_{m}) \\ − & r(1, d_{2}, ..., d_{m}) = 1 + r(d_{2}, ..., d_{m}) \\ − & w(D) = max\{r(D − 1) − 1; e(D − 1, w(D − 1))\} \\ − & r(D) = 1 + ∑_{j=1}^{m} [\binom{d_{j} − 1 + w(D)}{d_{j} − 1} − 1] \end{matrix}

where the last two equations hold when d_{1} > 1.

The following is the main result of the paper: Theorem 2. Suppose 1 ≤ d_{1} ≤ ... ≤ d_{m}. Set D = (d_{1}, ..., d_{m}). Then the integer m(D) defined by the equation

m(D) = max{e(D, w(D)), r(D)}

is such that for every r ≥ m(D), the generic V^{(r,D)} is unirational.

Remark: The word “generic” can be made precise: we say that there exists a dense open set U in Int(r, D) with the property that the fiber of the projection Int(r, D) → Int(r, D), over each point of U, is unirational.

The proof of theorem 2 is based on the following Proposition. Set D = (d_{1}, ..., d_{m}), where 1 ≤ d_{1} ≤ ... ≤ d_{m}. Then for every integer r ≥ r(D) there exists an integer s and a rational dominant map ϕ such that the diagram

P_{k}^{s} × Rey(r, D, w(D)) →^{ϕ} Rey(r, D, w(D))
↘  ↓
Rey(r, D, w(D))

commutes.

Proof of the proposition: If d_{1} = 1, we note that there are obvious isomorphisms which make the following diagram commute

Gr(r − 1, r) × Int(r − 1, d_{2}, ..., d_{m}) → Int(r, 1, d_{2}, ..., d_{m})
↓  ↓
Gr(r − 1, r) × Int(r − 1, d_{2}, ..., d_{m}) → Int(r, 1, d_{2}, ..., d_{m})

So we may assume d_{1} > 1 in the following.

We shall argue by induction on n = max{d_{1}, ..., d_{m}}.

-Starting point of induction Suppose m = 1 and d_{1} = 2. In this case the claim of the proposition can be stated as follows : “for any given pair (quadric,k-point on the quadric) we may choose a canonical birational isomorphism with P_{k}^{r−1}”. Of course this is well-known, and can be gotten e.g. fixing an isomorphism of P_{k}^{r−1} with a hyperplane of P_{k}^{r} and projecting from the chosen point of the quadric.

-Inductive Step Suppose n = d_{m} > 2 or m > 1. Let q be any integer smaller than r − m (the dimension of the fiber of Int(r, D) → Int(r, D)). Towards the end we will have to add some further restrictions on q, but they are not indispensable at the present stage of the proof. We build the family L of the q-dimensional linear subspaces of P_{k}^{r} which appear in the Reye congruence :

L  → Gr(q, r)
↓  ↓
Rey(r, D, q) →^{π} Gr(q, r)

(L is the fiber product in the diagram, and clearly L ⊂ Rey). Next we subtract L from Rey : F = Rey − L. There is a natural projection

\begin{matrix}F \\ ↓ \\ Rey(r, D, q) \end{matrix}

Now we choose a birational map β that makes the following diagram commute

P_{k}^{q} × Rey →^{β} L ↪^{j} Rey
↓  ↓  ↓
Rey  →^{1} Rey →^{1} Rey

and such that for each ζ in a dense open set of Rey the map β_{ζ} be linear and surjective over L_{ζ}. It is easily seen that β exists. We write ψ = j ∘ β. Fix a linear space N ⊂ P_{k}^{r} of dimension r − q − 1. Fix P_{k}^{q} ⊂ P_{k}^{q+1} and ξ_{0} ∈ P_{k}^{q+1} − P_{k}^{q}. There exists a dense open set U ⊆ Rey such that if (V, ℓ) ∈ U, then N ∩ ℓ = ∅. We claim that there exists a dominant rational map χ : N × P_{k}^{q+1} × Rey → P_{k}^{r} × Rey such that for every η ∈ N, and for a generic ζ ∈ Rey the map χ_{(η,ζ)} is linear and makes the following diagram commute:

(η, ξ_{0}, ζ)  ⟶ (η, ζ)
↓  ↓
η × P_{k}^{q+1} × ζ ⟶^{χ_{(η,ζ)}} P_{k}^{r} × ζ
↑  ↑
P_{k}^{q} × ζ  ⟶^{ψ_{ζ}} P_{k}^{r} × ζ

Now we get the family F → N × Rey(r, D, q) as the fiber product in the following diagram :

F  →  F
↓  ↓
N × P_{k}^{q+1} × Rey(r, D, q) → P_{k}^{r} × Rey(r, D, q)
↓
N × Rey(r, D, q)

Each fiber of the family F → N × Rey(r, D, q) comes from the intersection of a V − ℓ with a q + 1-dimensional linear space η ∗ ℓ (the join of η and ℓ). Take the closure of F in N × P_{k}^{q+1} × Rey(r, D, q) and denote it by F again. By the choice of dimensions, this is a family of complete intersections with frame, of type V^{(q+1,D−1)}.(This is only a matter of performing some calculations with the equation of a generic V). So there exists a rational map τ that make commute the diagram

F  → Int(q + 1, D − 1)
↓  ↓
N × Rey →^{τ} Int(q + 1, D − 1)

We have the Lemma. τ is dominant. Proof: In fact, choose ξ = (V, ℓ) with V ∈ Int and ℓ ∈ Gr and a set of coordinates (x, y) for P_{k}^{r} such that

\begin{aligned}ℓ & = \{(x, y)|x = 0\} \\ N & = \{(x, y)|y = 0\} \end{aligned}

The V^{(r,d_{1},...,d_{m})} is defined by a set of m equations of the form

g_{i}(x, y) = ∑_{j=1}^{r−k−1} x_{j}g_{ij}(y) + G_{i}(x, y) = 0  i = 1, ..., m

where the g_{ij}’s are homogeneous polinomials of degree d_{i} − 1 in the variables x and the G_{i}’s are homogeneous polinomials of degree d_{i} in the variables x, y, which belong to square of the ideal (x_{1}, ..., x_{r−k−1}). Then, τ(η, ξ) is the complete intersection that result taking the closure in η ∗ ℓ of the variety (V − ℓ) ∩ (η ∗ ℓ). Fix the isomorphism δ : P_{k}^{q+1} → η ∗ ℓ given by (z, y) ↦ (zη, y). Then the equations of τ(η, ξ) are

g′_{i}(z, y) = ∑_{j=1}^{r−k−1} η_{j}g_{ij}(y) + z^{−1}G_{i}(zη, y)  i = 1, ..., m.

Now, claiming that τ is dominant is equivalent to saying that the generic polynomial g′_{i} comes from the generic polynomial g_{i} by means of the substitution δ, and this is clear.

Next we get the family F′ from the diagram

F′  ↪  F
↓  ↓
N × P_{k}^{q} × Rey ↪ N × P_{k}^{q+1} × Rey

Again, F′ is a family of complete intersections with frame, of type V^{(q,D−1)}. Let σ : N × Rey(r, D, q) → Int(q, D − 1) the map which plays the role of τ in the analogous diagram for F′.

Suppose now that dim N > dim Int(q, D − 1). Under this assumption, we show that for every ξ in a dense open subset U ⊆ Rey(r, D, q), the map σ turns N into a vector bundle over Int(q, D − 1). In fact, keeping the notation of the lemma, by straigthforward calculations we get the equations of the complete intersection parametrized by σ(η, ξ) for η = (0, ..., 0, η_{q+1}, ..., η_{r}) ∈ N :

∑_{j=q+1}^{r} η_{j}g_{ij}(x_{0}, ..., x_{q}) = 0  i = 1, ..., m

Now, each choice of values for the η_{j}’s determines a point in Int(q, D − 1), by the the above formulas. Viceversa, fixing a point in Int(q, D − 1), we may look at the above formulas as a system of linear inhomogeneous equations in the variables η_{j} . If this system of equations has maximal rank (this happens for the ξ’s in a dense open subset of Rey), the fibers of σ_{ξ} over each point in Int(q, D − 1) form a linear subvariety of N, thus giving to N the structure of a (projectivized) vector bundle over Int(q, D − 1).

Here some remarks are in order : σ_{ξ} maps the projective space N to the product of projective spaces Int(q, D − 1) ≃ P_{k}^{N(q,d_{1}−1)} × ... × P_{k}^{N(q,d_{m}−1)}. If we compose σ_{ξ} with the projection π_{i} onto one of the factor, we get a linear map. We express this by saying that σ_{ξ} is a linear rational map. If we dehomogenize with respect to suitable coordinates in N and in the P_{k}^{N(q,d_{i}−1)}, we get a linear map between linear spaces:

σ_{ξ} : A^{r−k−1} → A^{N(q,d_{1}−1)} × ... × A^{N(q,d_{m}−1)}.

In this way, the (closure of) the image of σ_{ξ} turns into a linear subspace of the codomain. In general, we say that a subvariety of Int is linear if it is birational to a linear subspace of A^{N(q,d_{1})} × ... × A^{N(q,d_{m}−1)} by means of a suitable dehomogenization. We will use liberally this terminology, carrying it on to introduce a Grassman variety of the linear subvarieties of Int (definition left to the reader), and so on, when we will need further machinery. In effect all this works only at the birational level, that is for a dense open set of N and a dense open set of Int(q, D − 1), and one should be convinced that this is all what we need.

Let’s get back to the proof : we have examined the case N ≥ dim Int(q, D−1). On the other hand, if the dimension of N is smaller than or equal to the dimension of Int(q, D − 1), the same argument shows that σ is a birational isomorphism from N onto a linear subvariety (in the sense we’ve just defined) of Int(q, D − 1). We work out this case as follows : let c be the codimension of σ(N) in Int(q, D− 1). First, note that if L is the dual of the vector space k^{q+1}, then Int(q, D − 1) = P(Sym^{d_{1}}(L)) × ... × P(Sym^{d_{m}}(L)). Consequently, the natural action of GL(q + 1) on k^{q+1} induces a linear action of P GL(q) on Int(q, D − 1). Assume that c ≤ q. Then the group G_{m}^{c} is naturally embedded in P GL(q) as the subgroup of those trasformations that act as dilations for the coordinates x_{1}, ..., x_{c} and leave invariant the other coordinates. Let ξ ∈ Int(q, D − 1) and let

∑_{|I|=d_{m}−1} a_{I}x^{I} = 0

be an equation of the m-th hypersurface which defines ξ, in a fixed coordinate system x_{0}, ..., x_{q}. We denote by \overline{Int} the subspace consisting of the ξ’s such that the coefficients a_{I} of the monomials x_{i}x_{0}^{d_{m}−2} for 0 ≤ i ≤ c, are all equal. It is quite clear that the quotient Int/G_{m}^{c} is birational to \overline{Int}. More precisely, we have

Int(q, D − 1) ≃ G_{m}^{c} × \overline{Int}(q, D − 1).

Summarizing, Int→^{p}Int/G_{m}^{c} turns into a fiber bundle with a rational section \overline{Int} that is a linear subspace. It follows that every linear subspace in a neighborood (in the grassmaniann topology) of \overline{Int} is again a section of the fiber bundle. Now it is easy to check that for all the ξ’s in a dense open subset of Rey(r, D, q), the image σ(N) is transversal to the projection p, that is, the image is a section. We build α as the pull-back

M(r, D, q, w)  →^{α} N × Rey(r, D, q)
↓  ↓^{p∘σ}
Rey(q, D − 1, w) →^{p∘π}  \overline{Int}(q, D − 1)

where M(r, D, q, w) = (N × Rey(r, D, q)) ×\overline{Int} Rey(q, D − 1, w) is the fiber product in the diagram and w = w(D − 1) is the integer given by induction. The above arguments lead us to the following conclusions :

1) if the codimension of σ(N × ξ) is smaller than or equal to q, the map (p ∘ σ)_{ξ} turns N × ξ into a vector bundle over \overline{Int}(q, D − 1), for all the ξ in an open dense subset U ⊆ Rey(r, D, q); in particular, it is easy to check that there exists a birational isomorphism (N × U) ×\overline{Int} ×Rey(q, D − 1, w) ≃ P_{k}^{n} × Rey(q, D − 1, w) × U for the appropriate fiber dimension n; 2) if q ≥ e(D − 1, w) then p ∘ π is dominant and in turn, α is dominant;

3) if 1) and 2) are filled, then for ξ ∈ U the fiber product (N ×ξ)×\overline{Int}Rey(q, D− 1, w) is a vector bundle over Rey(q, D−1, w), in particular it is rational. (Here we omit the verification that Rey is rational, that is quite obvious). The inclusion P_{k}^{q} ⊂ P_{k}^{q+1} that we fixed above gives rise to a rational map

(N × Rey(r, D, q)) ×\overline{Int} Rey(q, D − 1, w)→^{γ}Rey(q + 1, D − 1, w)

in this way : if ((η, ξ), (V, ℓ)) is in the domain, then there is a unique T ∈ G_{m}^{c} with the property that σξ = T V. We write T = \frac{σξ}{V} and define γ as follows

((η, ξ), (V, ℓ)) ↦ (τ(η, ξ), \frac{σξ}{V} ℓ).

We build J and F as fiber products in the diagram

F  → Rey(q + 1, D − 1, w)
↙  ↓  ↓
F  J  →^{λ} Rey(q + 1, D − 1, w)
↓  ↙  ↙
N × Rey(r, D, q) →^{τ} Int(q + 1, D − 1)

where λ is dominant since τ is such. The maps α and γ give a rational morphism

M(r, D, q, w)→^{δ}J.

Assume now that q + 1 ≥ r(D − 1). Then by induction there is a rational dominant map

ϕ : P_{k}^{s} × Rey(q + 1, D − 1, w) → Rey(q + 1, D − 1, w).

Since λ is dominant, we can transfer ϕ on F :

P_{k}^{s} × J → P_{k}^{s} × Rey(q + 1, D − 1, w)
↓  ↓^{ϕ}
F  →  Rey(q + 1, D − 1, w)
↓  ↓
J  →  Rey(q + 1, D − 1, w)

¿From conclusion 1) above, we deduce that there exists a birational isomorphism

P_{k}^{s} × M(r, D, q, w) ≃ {P_{k}^{s} × P_{k}^{n} × Rey(q, D − 1, w)} × Rey(r, D, q).

The term in brackets on the right-hand side is rational, and we will denote it simply by P_{k}^{t}, where t is the appropriate dimension.

At last, we obtain the following diagram :

P_{k}^{t} ×_{k} Rey(r, D, q) → P_{k}^{s} ×_{k} J
|  ↓
|  F
↓  ↓
M(r, D, q, w)  →^{δ}  J

from which, by composition we get the expected ϕ :

P_{k}^{t} ×_{k} Rey(r, D, q) → P_{k}^{s} ×_{k} J → F → F → Rey(r, D, q)

5. The dimension of N.

Now we discusss the conditions to be put on the dimension of N and on the other parameters involved in the determination of the lower bound for r(D). First of all, it is clear that w(D) = q. By looking at the assumptions that we made along the proof, we have two other conditions :

\begin{aligned}q + 1 & ≥ r(D − 1) \\ q & ≥ e(D − 1, w(D − 1)) \end{aligned}

It is also necessary to impose an upper bound on the codimension c of σ(N) in Int(q, D − 1). This translates in the following inequality :

dimN ≥ ∑_{j=1}^{m} \{\binom{d_{j} + q − 1}{q} − 1\} − q.

Obviously we have dimN = r(D) − q − 1 and this ends the proof of the proposition.

We can finally give the proof of theorem 2: Proof of theorem 2: From the proposition we see that if r ≥ r(D), then the generic V^{(r,D)} containing a linear variety of dimension w(D) is unirational. On the other hand, if r ≥ e(D, w(D)), then every V^{(r,D)} contains such a linear variety. The theorem readily follows.

6. References.

[1] U.Morin, Sull’unirazionalità dell’ipersuperficie algebrica di qualunque ordine e dimensione sufficientemente alta. Atti Congr. Un. Mat. Ital., Bologna (1940), pp. 298-302.

[2] A.Predonzan, Sull’unirazionalità della varietà intersezione completa di più forme. Rend Sem. Mat. Padova, 18 (1949), pp. 161-176.

[3] C.Ciliberto, Osservazioni su alcuni classici teoremi di unirazionalità per ipersuperficie e complete intersezioni algebriche proiettive. Ricerche di Mat., Napoli, vol. XXIX, fascicolo 2 (1980), pp. 175-191.

[4] J.P.Murre, Discussion of a theorem of Morin. Proceeding of a seminar on ”Argomenti di Geometria Algebrica”, Povo (Trento) (1979) cyclostyle notes.

[5] A. Predonzan, Intorno agli S_{k} giacenti sulla varietà intersezione completa di più forme. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 5 (1948), pp. 238-242.

[6] L.Ramero, Problemi di Razionalità. Università di Pisa, Thesis, Pisa (1989).

Address of the Author.

Lorenzo Ramero Scuola Normale Superiore 56100 Pisa Italy.