Definability over subfields of rational maps to projective space: Cremona transformations, degree, divisors

Question

Recently I was led to consider some questions about Cremona transformations over arbitrary fields, which led deeper than planned into parts of algebraic geometry. I studied only algebraic groups in grad school, and read more by myself over the years with no intention of making this a focus, so my expertise is limited. Most of all I am trying to locate references to a few results which seem fundamental. But, to my surprise, diligent searches through older (Hodge and Pedoe, Semple and Roth, surveys edited by Snyder,...) and many later (mid 50s on) treatises, my favorites being more recent ones by Liu and Vakil, failed to turn up anything relevant, even in char 0.

Are the following obvious or well known -- and WHERE are they?:

(1) Any rational map $\phi: P^n_k \dashrightarrow P^n_k$ that becomes birational on extending to a larger field is already birational over k. In other words, a Cremona transformation and its inverse are always defined over the same subfields.

One way, maybe not the most direct, uses explicit inversion formulas for coeffs. It turns out that for any given degree (as below), independently of the field, there are a finite number of possible formulas. This simplifies and goes well beyond a result of Semple and Tyrrell (1968) on types over fields containing C. I am writing a short article about these ideas.

Somewhat related, but almost trivial (why is this not in standard texts?), is:

(2) The degree d of any $\phi: P^m_k \dashrightarrow P^n_k$ does not change under field extension.

By degree it is meant the least degree d of homogeneous polynomials that represent phi. This is not to be confused with a more common notion of degree, say for certain finite dominant morphisms, which often conflicts with the degree used here, e.g. when $\phi$ is birational. Equivalently, use the least d such that $\phi$ factors as the d-fold (Veronese) embedding of $P^m_k$ followed by a projection.

In terms of divisors, the problem is to show that over an algebraic closure K of k, the largest $D$ such that $\phi_K$ factors through $\mathcal L(-D)$ is definable over k. This has some connection with results of Chow (1950) obtained via Chow coordinates. There may be more relevant ideas in Weil's Foundations, which I found indigestible. Ignore this if you are not interested in history.

In terms of twisting sheaves, d seems to be the least number such that the image of $\mathcal O(1)$ under $\phi^*$ is embeddable in $\mathcal O(d)$. I have been told that using $\phi^*$ it is obvious that d does not change under field extension, but I did not get any extra insight into this by thinking in terms of Proj and sections of sheaves, even after consulting good sources such as Vakil's latest version.

(3) Is there a fairly simple proof along these lines, and what would be the key points?

The only way I know is to abandon sophisticated points of view and use a basic result whose origin/locations remain bafflingly elusive:

(4) If some multivariate polynomials over k have no common factor (of positive degree), the same is true after extending the field. [WHERE is this?]

This is of course easy (reduce to studying the effect on factorization of polynomials when a generator is added to the underlying field; there are 3 cases), and should appear in some older algebra book, any time after Steinitz (1910) on field theory. I am only mildly interested in refs to constructive proofs, which may be much older (over C) but would be more complicated and presumably use some sort of resultant.

Going beyond what motivated me, but obviously a natural question, is:

(5) Are there similar results with the domain replaced by something more general than an open set (the base-point-free locus of $\phi$) in the variety $P^m_k$?

Answer 1

(1) The degree of a dominant generically finite rational map doesn't change after extension. If $f : X \dashrightarrow Y$ is a such rational map of geometrically integral varieties (e.g. $\mathbb P^n$) over $k$, then the field of function fields of $X_K$ (resp. $Y_K$) is $k(X)\otimes_k K$ (resp. $k(Y)\otimes_k K$). Of course, this operation doesn't change the degree $[k(X) : k(Y)]$.

(2)-(3). Let $U$ be a non-empty open subset where $\phi$ is defined. Then $\phi^{*}O_{P^n}(1)$ is an invertible sheaf on $U$ which extends uniquely to an invertible sheaf $L$ on $P^m_k$ (because $P^n_k$ is regular and we can extend Weil divisors on $U$ to $P^n_k$). But then $L$ is isomorphic to $O_{P^m}(d)$ for a unique $d$, which is also your 'degree of $\phi$' and we have $\phi^{*}O_{P^n}(1)\simeq O_{P^m}(d)|_U$. This property is clearly unchanged by field extensions.

(4) Yes, you are just saying that the intersection of some hypersurfaces has codimension at least $2$. But the codimension of a subvariety is invariant by field extension.

(5) The only problem I see is to define the degree of $\phi$ is the sense of (2).

Answer 2

Thanks for the quick reply! (1) was too easy; I should have thought more. Also my remarks about divisors were not properly stated. The main point, which with great respect I don't believe has been answered correctly, is that the degree of rational maps between projective spaces, as defined in my post and used in some articles (others call it order, another overworked name) does not seem to fall out so easily from sheaf theory as may be widely believed.

When extending $\phi^{*}\mathcal O_{P^n}(1)$ to the base locus $B$, it seems to me from a detailed working out presented below that the stalks above $B$ of the Module vanish, and I do not see how extending or pulling back any kind of divisors could suggest a different conclusion.

Start with a dominant rational map $\phi : X \dashrightarrow Y$, where $X=P^n_k$ and $Y$ is a closed subvariety of $P^n_k$, with notation like $y_i=h_i(x)$, where the $h_i$ are homogeneous polynomials of degree d in the $x_j$ that define $\phi$.
Suppose that, over $k$, the $h_i$ have no common factor. The aim (admittedly artificial, in view of the almost trivial (4) of my note) is to use sheaf theory to determine this $d$ from $\phi$ alone, and especially to see that it cannot decrease if the field is extended.

Only quotients of degree 0 are directly encountered. For example, working locally, say with stalks, there is a (coordinate-based) definition of each module of $\mathcal O_Y(1)$ in terms of globally defined generators that (to emphasize their abstract nature) will be called $g_0, .. , g_n$ rather than, say, $y_0, \dots, y_n$. The relations are those of the form $g_i = y_i/y_j . g_j$ that make sense ($y_j \neq 0$ at that point in $Y$). After applying $\phi^*$, the module above a point $x \in X$ can be obtained by extending the scalars (tensoring) from one local ring to another, in effect keeping the same generators $g_i$ and the same relations $g_i = h_i/h_j . g_j$, now regarding k(Y) as a subfield of k(X).

The induced sheaf on $X$ is isomorphic to a subModule of $\mathcal O_X(d)$, where $g_i$ maps to the element usually identified with the polynomial $h_i$. Less clear if we refuse to use (4) is that, even after extending the field, no smaller $d$ will work, but let's continue.

Each abstract $g_i$ gives a global section that vanishes on the zero set of $h_i$ since it vanishes whenever $h_i = 0$ but some $h_j \neq 0$. Thus, above any point in the base locus $B$ the module is 0, so the sheaf is not invertible if base points exist. There no longer seems to be an obvious way to proceed with these ideas.

To conclude, I believe (4) is essential for defining the degree of rational maps, and I am astonished not to have found any source mentioning something so basic. It is too elementary to merit a proof via algebraic geometry, which when given in detail would I believe involve working with a polynomial ideal (the variety may be empty), and studying quotients by prime ideals above this ideal to look at transcendence degrees of function fields. I will post in other places, say forums on commutative algebra, to try to track down good references to (4).

Answer 3

This is a response to the last comment, but I don´t know how else to post it (too long).

The above explains well what happens over algebraically closed fields. Pulling back hyperplanes gives a a unique divisor class dH, hence d, which it seems will not change under field extension (see below for a detail).

This is the sort of divisor theory I am trying to unlearn, which led to my posts, and reading Liu´s book would be one of the best ways to proceed. I hope he won´t mind some statements about what goes wrong with divisors over arbitrary fields, which is presumably one more reason (after base change) why the sheaf approach is so prevalent.

With $\phi$ and the $h_i$ as before, look at the zero set of a $k$-linear combination of the $h_i$, or of an irreducible factor of it. Is it defined by a single equation? Think for example of sections that are elliptic curves, restricted to small fields $k$. How then would we define $d$ over $k$? What happens when the field is extended?

Supposing a sufficiently nice situation so that such problems can be ignored, when the field is extended the section still gives the class dH, but there is still the strange question about whether $d$ could decrease, which would happen only if the $n+1$ equations used to define the base locus (a possibly empty set) define over $K$ a variety of codimension 1. This is ruled out by the almost trivial (4), or via more sophisticated ideas about dimension such as heights of prime ideas, as in Liu´s book, or transcendence degrees.

(4) does in fact have something to do with divisors, so perhaps I should write more than in my original post (not having seen this anywhere). If we think of divisors on projective space via homogeneous rational functions rather than hypersurfaces, not worrying about geometry nor 0-gradings to get partial functions, it always makes sense to extend from k to K and say what is meant for a divisor over $K$ to be $k$-rational. Given $\phi$, we can only work directly with functions of quotients $h_i/h_j$, and can in the usual way define the least divisor $D = D_\phi$ such that $\phi$ factors through $\mathcal L(D)$, then ask whether this $D$ could change under field extension. The fact that is does not is just a translation of (4). In other words, if a rational map $\phi$ between projective spaces over $K$ is definable over a subfield $k$, the same is true of $D_\phi$. This divisor turns out to be the least common multiple of the $h_i$, assuming the gcd of these is 1.

No doubt all this is well-known, but what references are there?

Answer 4

Sorry for my comment based on thinking of Weil divisors only in terms of closed points, something I have still not completely unlearned. In the Weil divisor approach one runs naturally into (4) but it can be avoided at the cost of doing something a little deeper (see earlier remark about a strange question). There are no real mathematical issues, just a matter of taste about how to proceed.

To illustrate, suppose homogeneous polynomials $g_i$ of the same degree represent $\phi: P^n_k \to Y \subset P^m_k$, with $Y$ not in a hyperplane. Then $y_0$ on $P^m_k$ induces an effective Weil divisor on $P^n_k$. This is basically a polynomial, up to scalars. Is it $g_0$? In general, NO! It is $g_0/g$ that will give $d$, where $g$ is the gcd of the g_i. Under field extension does the gcd remain the same or can $d$ decrease? This is just (4).

Of course (4) is no more than an easy exercise with polynomials and fields, based on adjoining an element that is either transcendent, separable, or (say) a $p^{\text th}$ root. There are several slightly deeper prooofs, including a slick one by flatness. What I find astonishing is that despite its relevance it seems to have gone unnoticed. While still searching in many likely places, I have so far found NOTHING.

As a more didactic point, via (4) one needs little machinery to obtain a result on the field-independence of factoring rational maps on $P^n_k$ through Veronese embeddings, and to see that if a rational map is defined over $k$, so is its associated divisor. This does not come close to Chow´s result that there is a well-defined smallest field of definition of a divisor, which may be properly contained in $k$. Presumaby there are now more modern approaches than the original via Chow coordinates.

I would like to close by thanking Prof. Qing Liu for taking the time to help clarify the ideas involved.