Rational maps and Cremona transformations A primer to their ideal theoretic side Aron Simis
Abstract Notes for talks at Purdue University (Spring 2008), where one focus on a leisure introduction to recent methods in the ideal theory of rational and birational maps.
Introduction and terminology Let k be an arbitrary field . For the purpose of the full geometric picture we may have to assume k to be algebraically closed. However k will have arbitrary characteristic except for a few clearly marked passages. We denote by Pn = Pnk the nth projective space with homogeneous coordinates (a0 : · · · : an ) over k. Recall that Pn is the quotient space of An+1 \ {0} by the homothetical action. Let X ⊂ Pn be a projective subvariety, which will be assumed to be integral (i.e., reduced and irreducible or, equivalently, its largest defining ideal I(X) ⊂ k[x] = k[x0 , . . . , xn ] is prime). Thus, X admits a function field k(X) and one has the main dimension theorem of these preliminaries, namely, dim X = tr.deg.k k(X), where dim X is a combinatorial-like dimension defined in terms of lengths of chains of subvarieties. Let Pm be yet another projective space. We wish to consider rational maps with “domain” X and target Pm . The quotation marks indicate that there will be some difficulties as to what will be exactly the domain of these maps. The original thinking about rational maps had a great deal of vagueness. Clarity only came to grip under Zariski’s vast revision of the subject in completely algebraic terms and this is how we think of it till these days. Of course the terminology comes from having these maps defined in terms of rational functions on the domain, i.e., fractions of polynomial functions thereof. As is known, there are two different fields around in the theory of projective varieties: the “big” field of fractions of the homogeneous coordinate ring k[X] = k[x]/I(X) of X and the “small” field of fractions of an affine piece of X. While the first gives perfectly defined regular functions on the structural affine cone over X, it is the latter that is called the function field of X because it makes sense to say for its elements that they induce locally well-defined functions on X - while the elements of k[X] do not. The notation k(X) for the function field is highly incoherent with the previous notation k[X] but we will leave it at that since the go-between the two fields is well-known from elementary courses. Definition 0.1 A rational datum of X to Pm is a collection of elements f1 /f0 , . . . , fm /f0 ∈ k(X), where f0 , f1 , . . . , fm ∈ k[X] are forms of the same degree, with f0 ̸= 0. In a slightly imprecise way, the above datum induces a map with target Am ⊂ Pm and domain a certain subset of closed points of X, namely, given p = (a0 : · · · : an ) ∈ X, with ai ∈ k, such that f0 (p) = f (a0 , · · · , an )) ̸= 0, the image of p is the point with affine 1
coordinates (f1 (p)/f0 (p), . . . , fm (p)/f0 (p)) ∈ Am k . If one thinks of points in the language of Weil’s school, P = (¯ x0 : · · · : x ¯n ) can be thought of as being the generic point of X, where x ¯i denotes the residue of the variable xi in k[X]. In this version, the image of the generic point is the “affine point” (f1 (P )/f0 (P ), . . . , fm (P )/f0 (P )) = (f1 /f0 , . . . , fm /f0 ). Thus the coordinates of the image of the generic point of X generate a k-subfield of k(X) and can be thought of as the generic point of a subvariety whose affine coordinate ring is k[f1 /f0 , . . . , fm /f0 ]. The map thus obtained is called a rational map – hence the usage of the notation F : X 99K Pm to indicate such a map, where the dotted tail of the arrow reminds us of the partial domain of the map. We next highlight the main features of the concept. • The finitely generated k-domain k[f1 /f0 , . . . , fm /f0 ] ⊂ k(f1 /f0 , . . . , fm /f0 ) ⊂ k(X) is the affine coordinate ring of a uniquely defined subvariety of Am ⊂ Pm . The closure of the latter Y ⊂ Pm is called the image of the rational map F . By abuse, we also say that the rational map F is defined by the homogeneous datum f0 , f1 , . . . , fm and frequently use the notation F = (f0 : f1 : . . . : fm ) which as we will indicate below conveys the lack of uniqueness of choosing these forms. • We emphasize en passant that the image of a rational map on an integral projective subvariety is an integral projective subvariety of the target. As such, its homogeneous coordinate ring is isomorphic to the subring k[f0 , . . . , fm ] ⊂ k[X] as graded k-algebras - we say that one obtains the first from the second by a degree renormalization. Thus, although the dealing with rational maps is rather subtle, the image of such maps is a rather easily understood object in algebraic terms. • One says that F is birational onto its image if there exists a rational map G: Y 99K Pn whose image is X such that F and G are inverses to each other as field maps, i.e., if they define a k-isomorphism of fields k(X) ≃ k(Y ) and its inverse. • One can also express the latter behavior at the level of the homogeneous data, namely, if F = (f0 : f1 : . . . : fm ) and G = (g0 : g1 : . . . : gn ) then F is birational onto its image with inverse map G if and only if the following relations are satisfied: (f0 (g0 , g1 , . . . , gn ): . . . : fm (g0 , g1 , . . . , gn )) = (y¯0 : . . . : y¯m ) and (g0 (f0 , f1 , . . . , fm ): . . . : gn (f0 , f1 , . . . , fm )) ≡ (x¯0 : . . . : x¯n ) n as tuples of homogeneous coordinates in Pm K(Y ) (resp. PK(X) ) where K(Y ) is the field of fractions of k[Y ] (resp. of k[X]).
• If d is the common degree of f0 , f1 , . . . , fm , it is not difficult to see that this is the case if and only if the domains k[f0 , f1 , . . . , fm ] ⊂ k[k[X]d ] = k[(x)d ]/I(X)d have the same field of fractions. • There is nothing unique about the choice of f0 , f1 , . . . , fm , as one learns quite early in a first course. Nevertheless, any other choice of such a representative of the map 2
′ , necessarily satisfies the condition that the 2 × 2 minors of the F , say, f0′ , f1′ , . . . , fm matrix ( ) f0 f1 . . . fm ′ f0′ f1′ . . . fm
vanish (as elements of the homogeneous coordinate ring k[X] of X). Trivial remark as it is it represents a notable clarification in the theory. I believe it first appears in a paper of Zariski in an issue of the American J. Math. from the early nineteen forties which I would presently not be able to quote precisely. • Note that the latter condition also means that the above matrix has rank 1 over the field of fractions K(X) of k[X], i.e., the two row vectors are proportional over this field. This justifies the previous notation F = (f0 : · · · : fm ) (rigorously, an element of Pm K(X) ). • When considering the base locus of a rational map there is a potential confusion as to whether one is looking at the projective scheme defined by the forms f or more accurately at the homogeneous ideal generated by these forms – of course, for the set theoretic version of this locus one may as well go all the way to the radical of the ideal. However, in the present approach we will need to be fairly precise about the ideal of defining equations of the associated Rees algebra of the ideal version. Moreover, since saturation is a truly ideal theoretically operation one may loose track of the nature of the subalgebra k[f0 , . . . , fm ] which should be left intact as it defines the image of the given map. Also we will be using the subalgebra and the defining ring of the blowup on the same foot, hence one had better keep the Rees algebra intact as well. Quite a bit of the nature of the statements and their proofs appeal to the syzygies of an ideal representative of F , however the very format of the conditions do not directly involve them. This is because the condition for birationality rather requires dwelling on the “hidden side” of the Rees algebra, as will be made more precise. It is also remarkable, as we see it, that the theory is in terms of a certain jacobian matrix with respect to variables that only appear linearly in the pertinent equations - a so-called weak Jacobian dual matrix - hence there will be no restriction on the field characteristic, certainly a non-negligible point in birational geometry as is well-known.
1
Degree sequences
As a general proviso, the ideal generated by the r × r minors of a matrix φ will be denoted by Ir (φ). We are given an integral variety X ⊂ Pn , with homogeneous coordinate ring R = k[x]/I(X) = k[x0 , . . . , xn ]/I(X). If f = {f0 . . . , fm } are forms in R of the same degree d, we will consider the minimal free graded presentation of the homogeneous ideal (f ) ⊂ R. φ
⊕s R(−ds ) −→ Rm+1 (−d) → R
(1)
As is customary, we will by abuse call φ both the map above or a representative matrix thereof. 3
Proposition 1.1 Let F : X 99K Pm be a rational map and let f = {f0 . . . , fm } be a representative of F . Set I = (f ) ⊂ R. (i) The set of representatives of F correspond bijectively to the homogeneous vectors in the rank one graded R-module Hom(I, R) = ker (φt ), where t denotes transpose. In particular, F has a unique representative of lowest possible degree (up to elements of k\{0}) if and only if I has grade at least two – in which case every other representative is a multiple of the latter by a factor in R. (ii) (Assume that k is algebraically closed ) Let Jφ := I1 (ker (φt )) ⊂ R stand for the ideal generated by the entries of ker (φt ). Then the closed points of X at which F is welldefined correspond to the homogeneous sub-maximal ideals of R containing Jφ . In particular, F is a regular (i.e., everywhere defined) map if and only if Jφ is primary to the irrelevant ideal R+ . Proof. (i) Given another representative f ′ of F , then by definition ( ) f0 f1 . . . fn I2 = 0, f0′ f1′ . . . fn′ i.e., the two sets f and f ′ are proportional by a factor which is a homogeneous element of the field of fractions K of R. This establishes a bijection between the set of representatives of F and the set of homogeneous elements of K that drive I = (f ) into R. The latter generate the fractional ideal R:K I ≃ Hom(I, R). Finally, it is known that the the natural inclusion R ⊂ Hom(I, R) is an equality if and only if I has grade at least two. Therefore, F is uniquely represented up to proportionality with proportionality factor in k[X] if and only I has grade at least two. (ii) Clearly, ker (φt ) ≃ Hom(I, R)), so we pick up from there by using part (i). This means that any representative f = {f0 . . . , fm } of F corresponds to the unique homogeneous column vector ker (φt ) whose coordinates are f0 . . . , fm (we view the vectors of Rm+1 (−d) as column vectors). For any such f , let Z(f ) ⊂ . . , fm }. ∪ X denote the set∩of zeros of {f0 . ∑ Clearly, the map F is exactly defined in the set f (X \Z(f )) = X \ f Z(f ) = X \Z( f (f )), where f runs through the set of representatives of F (equivalently, the set of homogeneous vectors of ker (φt )) and (f ) ⊂ R denotes the ideal generated by the entries of the vector. Therefore, the domain of definition of F is precisely the complement of the set of zeros of the ideal of R generated by the coordinates of all homogeneous vectors of ker (φt ). Remark 1.2 The proposition emphasizes the effective side of determining whether a rational map F : X 99K Pm is regular. Thus, the ideal I1 (ker (φt )) could naturally be taken as definition of the base ideal of F though it is rather a loose ideal from the strict algebraic viewpoint. We emphasize that F being everywhere defined does not automatically imply the existence of a representative whose coordinates will generate a k[X]+ -primary ideal. The following simple example illustrates the order of ideas so far. Example 1.3 Let X ⊂ P2 (arbitrary characteristic) be the nodal cubic whose irreducible equation is y 2 z − x2 (x + z). The rational map (¯ x : y¯) : X 99K P1 is defined everywhere 4
except at (0 : 0 : 1): this is because the ideal of coordinates Jφ relative to the base ideal (¯ x, y¯) on k[x, y, z]/(y 2 z − x2 (x + z)) is (x, y) which vanishes exactly at (0 : 0 : 1). On the other hand, The rational map (¯ x2 : y¯2 ) : X 99K P1 is defined everywhere because the ideal of coordinates Jφ relative to the base ideal (¯ x2, y¯2) on k[x, y, z]/(y 2 z − x2 (x + z)) is (x, y, z). Nature is capricious: the first is a birational map onto P1 while the second one is a double covering map. Of course, we know that there could not be a rational inverse to the second map since the image is a smooth curve! Here is a side application that explains why any rational map defined on a smooth projective curve must be a regular map. Proposition 1.4 Let R be a 2-dimensional standard graded domain over an algebraically closed field k and let C = Proj(R) ⊂ Pn be any embedding of the corresponding curve. If R is regular locally in codimension one then every rational map with source C is regular. Proof. Let I ⊂ R denote the base ideal of the given rational map with source C. Let φ Rm −→ Rn → I → 0 be a free graded presentation of I as before and consider the dualized sequence 0 → I ∗ → Rm∗ → Rn∗ . Now map, say Rr , surjectively onto I ∗ so as to get the free exact sequence ψ
Rr −→ Rm∗ → Rn∗ of R-modules. Let P ⊂ R be any height one prime. Since RP is regular and rank(ψP ) = rank(ψ) = n − rank(φ) = 1 – hence ψP is injective – we get an exact sequence P 0 → RP −→ RPm ∗ → RPn ∗
ψ
of RP -modules. But again RP is regular of dimension 1, hence every RP -module – in particular, coker(RPm ∗ → RPn ∗ ) – has projective dimension at most 1. Therefore, the last sequence must split at its tail, i.e., I1 (ψ)P = I(ψP ) = RP . But P was arbitrary, hence I1 (ψ) is R+ -primary. Proposition 1.1 motivates the following notion. Definition 1.5 The degree sequence of the rational map F : X 99K Pm is the sequence d1 , . . . , dr (with d1 ≤ · · · ≤ dr ) of degrees of a minimal set of generators of Hom((f ), R), where f is a representative of F - or, equivalently, the sequence of non-increasing degrees of the column vectors of the syzygy matrix ker (φt ). For example, in both rational maps of Example 1.3 the degree sequence is 1, 2. Thus the knowledge of this sequence alone is not sufficient to detect the complete nature of the true locus where a rational map is defined. However, it is definitely an invariant of the rational map (i.e., does not depend on the chosen representative). In particular, if for one representative f the corresponding ideal has grade at least two – for example, if X = Pn – then every representative of F is a multiple of f by a factor which is a homogeneous element of k[X]. Clearly, this is then the unique (up to a nonzero factor in the base field k) representative with this property. This is also a condition under which the base ideal and the “true” base ideal coincide. In this case, the degree sequence consists of a unique number, namely, the degree of f . This degree will then be said to be the degree of F and the corresponding representative will generate the (uniquely defined) base ideal of F . 5
Example 1.6 Recall that there is a so-called Gauss map associated to any projective embedding X ⊂ Pn . This is defined in terms of the embedded projective tangent spaces to X at its smooth points, by associating to each such point the corresponding tangent space as an element of the Grassamannian G(d, n), where d = dim X. This map is not defined everwhere in general. If, moreover, one embeds the Grassmannian in a projective space Pm by, e.g., the Pl¨ ucker embedding, then the Gauss map becomes an example of a rational map. In the case where X is a hypersurface, defined by a form f ∈ k[x], one has G(n − 1, n) = Pn by the well-known point/hyperplane duality, hence it is not difficult to see that a representative of the Gauss map in this case is (∂f /∂x0 : · · · : ∂f /∂xn ) as viewed modulo (f ). Therefore the image of the Gauss map is actually the dual variety to the hypersurface. Corollary 1.7 (k perfect) Let X ⊂ Pn be a hypersurface of degree d ≥ 2. Then the degree sequence of the Gauss map F : X 99K Pn is d − 1 if and only if X is arithmetically normal. Proof. In this situation, the base ideal is Jf = (∂f /∂x0 , · · · , ∂f /∂xn )k[x]/(f ) – called the Jacobian ideal of k[x]/(f ). By the well-known Jacobian criterion, k[x]/(f ) is a normal ring if and only if Jf has codimension at least two.
2
Birationality in terms of Rees algebras
In this section we state a general criterion of birationality that encompasses previous similar results. The present format underwent several intermediate stages and special cases were brought up by previous authors, some of which are listed in the references. First we will state a general principle which brings Rees algebras into the picture. Here is a quick reminder of how Rees algebras and subrings naturally emerge from geometrical considerations related to resolving the locus of non-definition of a rational map. The ground geometry is actually simple to visualize in terms of the graph of the rational map: given F : X 99K Pm , the graph Γ(F ) of F is the closure in Pn ×k Pm of the ordinary graph of F as a regular map wherever it is defined – note that by Proposition 1.1, (ii), this locus is very precise. At least set-theoretically the image Y ⊂ Pm of F and the image of the projection of Γ(F ) to Pm are the same on a common suitable restriction. Likewise, it is set-theoretically apparent that if F is birational onto its image and G: Y 99K Pn is its inverse then the two graphs Γ(F ) and Γ(G) are equal on suitable restrictions of the two maps, hence must be equal. Of course, for this we are using at least twice (Zariski) topological arguments and a known dictionary from geometry to algebra. Now, going deeper one learns that if F = (f0 : · · · : fm ) and f = {f0 , . . . , fm } then Γ(F ) can be thought of as ProjX (⊕t≥0 I t ) where I is the sheaf theoretic version on X of the ideal (f ) ⊂ k[X]. So far we get a relative blowup gadget. However, we can also think of the closely related Spec (Rk[X] ((f ))) where Rk[X] ((f )) is the Rees algebra of the ideal (f ) ⊂ k[X]. If one observes that this ring is actually a bigraded k-algebra (with all ambient variables of degree 1) then it becomes slightly more precise to think of Γ(F ) as (bi)projective subvariety of (bi)projective space Pn ×k Pm . 6
The final goal is then to show how the geometric picture described above precisely translates in this language. This is the purpose of the next proposition, whose statement and proofs do not resort to (but could be translated back into) geometric arguments. Proposition 2.1 Let X ⊂ Pn and Y ⊂ Pm denote integral subvarieties of positive dimension and let F : X 99K Pm and G: Y 99K Pn stand for rational maps with image Y and X, respectively (so that, in particular, dim X = dim Y ). Fix sets of forms f = {f0 , . . . , fm } ⊂ k[X] and g = {g0 , . . . , gn } ⊂ k[Y ] which are representatives of F and G, respectively. The following are equivalent conditions: (i) F and G are inverse to each other. (ii) The identity map of k[X] ⊗k k[Y ] ≃ k[x, y]/(I(X), I(Y )) induces a bigraded isomorphism Rk[X] (f ) ≃ Rk[Y ] (g). Remark 2.2 As a preliminary, we note that the algebra Rk[X] (f ) (respectively, Rk[Y ] (g)) contains a copy of k[Y ] (respectively, of k[X]) as a k-subalgebra generated in bidegrees (0, 1) (respectively, in bidegrees (1, 0)). In particular both algebras are residue algebras of the standard bigraded k-algebra k[x, y]/(I(X), I(Y )). To see this, recall that Rk[X] (f ) ≃ k[X] [f t] ⊂ k[X][t]. Now, by assumption k[Y ] ≃ k[f ] ≃ k[f t] and these isomorphisms are homogeneous by an obvious degree normalization of f and of f t. Then the required identification of k[Y ] is with the second of these k-subalgebras. The argument for the other algebra is the same. Now, the above restatement in terms of Rees algebras is not completely satisfactory as it presumes that we are a priori given the coordinates of the candidate for the inverse rational map. In the next portion we will restate a more direct criterion which checks whether a given rational map is birational and, as a bonus, will allow to compute the inverse in terms of the given rational map only. The criterion depends on knowing some of the defining equations of a Rees algebra, but otherwise it is stated in simple linear algebra jargon. We believe it constitutes a reasonable step forward if confronted with the involved – sometimes obscure too – arguments in the classical authors.
3
Main criterion
We start by considering the bigrading of the Rees algebra RR ((f )) in more detail, where we set R := k[X] and f = {f0 , . . . , fm } is a representative of a rational map F : X 99K Pm . Let Jf ⊂ R[y] denote the kernel of the R-algebra homomorphism R[y] → RR (f ) mapping yj to fj - often called the presentation or defining ideal of RR (f ). This is a bihomogeneous ideal in the bigrading induced by the standard bigrading of R[y] = k[x, y]/I(X)k[x, y]. Thus, Jf is generated by biforms of various bidegrees (r, s), r ≥ 0, s ≥ 0. We will assume that f is set of minimal generators of (f ), so that Jf is actually generated in bidegrees (r, s), with s ≥ 1. Note that the minimal generators of Jf of bidegree (r, 1), with r ≥ 1, generate the defining ideal of the symmetric algebra SymR (f ). 7
Note that if f and f ′ denote representatives of the same rational map F : X 99K Pm then Jf = Jf ′ (so that RR (f ) ≃ RR (f ′ )). Therefore, we will denote this ideal simply by J . Now, fix a minimal set of biforms generating J and, among its constituents consider all those of bidegree (1, s), with s ≥ 1. These biforms generate a subideal J(1,∗) ⊂ J of the form I1 (ψ · (x)t ) ⊂ R[y], where ψ is a uniquely defined matrix with n + 1 columns and entries in k[y], namely, the jacobian matrix with respect to the variables x of the chosen elements lifted to k[x, y]. Clearly, ψ is a graded matrix since its rows are homogeneous vectors in k[y]. We will mainly consider ψ as a matrix over the homogeneous coordinate ring S = k[Y ] = k[y]/I(Y ) of the image Y ⊂ Pm . Note that ψ can be the zero matrix, as it happens in many cases. Definition 3.1 The matrix ψ above will be called a weak Jacobian dual matrix of the rational map F . We note its similarity with the main concept introduced in an earlier paper by Ulrich, Vasconcelos and this speaker (see [11]), on which it is inspired. The main feature that sets the present notion aside from the one in loc. cit. is that it is always defined while the latter required an a priori hypothesis for its existence. Remark 3.2 Though the definition of a weak Jacobian dual matrix depends only on F (and not on a representative of F ), it is slightly unstable as it depends on the choice of minimal bihomogeneous generators of J(1,∗) ⊂ J . However (just like in the ordinary homogeneous situation), for a fixed bidegree (1, s) the number of (1, s)-biforms in any minimal set of generating biforms of J ⊂ R[y] is invariant and equals the dimension of the correspondingly spanned k-vector subspace of the whole space of (1, s)-biforms of J . This implies that any two weak Jacobian dual matrices of the same rational map F will have the same sizes. Moreover, if F turns out to be birational then the next theorem implies that they also have the same rank over the homogeneous coordinate ring of the image of F . As a final matter of notation, if θ (resp. E) is a matrix (resp. a module over a ring A) then θt (resp. E ∗ ) denotes the transposed matrix (resp. the A-dual Hom(E, A)). Theorem 3.3 Let X ⊂ Pn denote an integral subvariety of positive dimension and let F : X 99K Pm stand for a rational map with image Y . The following are equivalent conditions: (i) F is birational onto Y (ii) dim k[X] = dim k[Y ] and for some (any) weak Jacobian dual matrix ψ one has rankk[Y] (ψ) = n and Im((ψ)t ) = Im(ψ)∗ . Moreover, when condition (ii) takes place then the transpose of any homogenous vector of kerk[Y ] (ψ) yields a representative of the inverse map. The proof relies heavily on Proposition 2.1. The following special case may be useful when depth k[Y ] ≥ 2. 8
Corollary 3.4 In the same setting and notation of Theorem 3.3, suppose that dim R = dim S and that F admits a weak Jacobian dual matrix ψ such that rankk[Y] (ψ) = n. If the grade of the ideal of n-minors In (ψ) ⊂ k[Y ] is at least two and cokerk[Y ] ((ψ)t ) is a torsionfree k[Y ]-module then F is birational onto Y . Proof. Set S = k[Y ]. By the previous theorem it suffices to show the equality Im((ψ)t ) = ψ
Im(ψ)∗ . For this, let K = coker(S n+1 −→ S p ). Dualizing, we get an exact sequence 0 → Im((ψ)t ) → Im(ψ)∗ → Ext1S (K, S) → 0. ρt
Now Im(ψ)∗ = ker ((S n+1 )∗ → (S q )∗ ) is a reflexive S-module being the kernel of a map between free modules over a domain (cf., e.g., [1, Proposition 16.34]). But Im((ψ)t ) is reflexive as well because by assumption coker((ψ)t ) is torsionfree. We claim that Ext1S (K, S) = 0. Indeed, the standing assumption that grade In (ψ) ≥ 2 implies that K is free locally in grade at most one, hence Ext1S (K, S) is zero locally in grade at most one. Therefore, grade Ext1S (K, S) ≥ 2. It is standard that the cokernel of an inclusion of reflexive modules vanishes if it has grade at least two. Thus, Ext1S (K, S) = 0, hence Im((ψ)t ) = Im(ψ)∗ , as required.
Illustrative examples The following examples are fairly simple, but may serve as a guide to the theory so far. While the first two can be checked via Corollary 3.4, the third one strictly requires the full power of Theorem 3.3. The three are examples with dim X = 1. The fourth example is in dimension two. Needless to say, in the examples we assume that the characteristic of the base field is sufficiently high with respect to the data. For critical characteristics the maps may not be birational - in accordance with well known examples - and the criterion shows where it fails. Moreover, even if the map is birational over a critical characteristic the inverse rational map may fail to have the generic degree sequence (i.e., the expected degrees coming from the high characteristic case) . The first two examples are special cases of the Gauss map or, taken more lightly, of dual varieties. Example 3.5 Let X ⊂ P2 be the nodal cubic f = x30 − x20 x2 + x21 x2 . Let F : P2 99K P2 be the rational map represented by f = {∂f /∂x0 , ∂f /∂x1 , ∂f /∂x2 } on k[x, y, z]/(f ). Then, as is well known, the image of F is the dual curve g to f . By a standard formula, the degree of g is 4. Since the previous criterion is effective, we can compute everything using, e.g., Macaulay. A computation gives y¯0 y¯1 y¯2 y¯12 y¯0 (¯ y1 − 3/2¯ y2 ) 3¯ y1 (¯ y0 − y¯2 ) ψ= 2 2 −9/2¯ y1 y¯2 y¯1 − y¯2 −4¯ y1 y¯2 2 2 2 2 −2¯ y1 3/4¯ y1 y¯2 y¯1 − y¯2 + 1/2¯ y0 y¯2 − 3¯ y2 whose kernel is generated by 4 vectors of standard degree 3. Thus, e.g., the inverse map is defined by the forms −¯ y1 y¯2 (4¯ y0 − 9/2¯ y2 )
y¯2 (3¯ y12 + y¯1 y¯2 − 3¯ y0 y¯2 ) y¯1 (¯ y02 − y¯12 − 3/2¯ y0 y¯2 ). 9
As it turns the true base locus of the inverse map coincides as a set with the singular locus of the dual quartic - this is of course as expected because of the well-known principle of reflexivity of dual varieties (characteristic zero). Yet another computation yields that ψ has the required properties for F to be birational and, moreover, the degree sequences of F and F −1 are, respectively, 2, 3 and 3, 3, 3, 3. It is also easy to verify by computation that the partial derivatives of g give a representative of F −1 . Moreover, neither rational map is regular, which can be readily effectively checked by Proposition 1.1, (ii) (cf. also Remark 1.2). Example 3.6 Let X ⊂ P2 be the elliptic cubic f = x30 + x31 − x32 . The computation is analogous and it turns out that the dual g is a sextic and the degree sequences of F and F −1 are, respectively, 2 and 4, 4, 4, 5. One of the available representatives in degree five is given by the partial derivatives of the sextic. Clearly, F is regular (no computation needed), while an effective computation shows that F −1 is not regular. As sets, the generalized base locus I1 (ker (ψ)) (cf. Remark 1.2) and the singular locus of g coincide, hence F −1 is not regular exactly at the singular points of g. The fact here that one can define the inverse map in terms of a representative of degree 4 – instead of the degree of the partial derivatives of the equation of the dual sextic – is at least curious and, to my knowledge, has not been pointed out before. Example 3.7 Let X = P1 and let F : P1 ≃ Y ⊂ P3 be the biregular parametrization of the non-normal rational quartic given by f = {x40 , x30 x1 , x0 x31 , x41 }. Since the coordinate ring S = k[Y ] has depth one, one cannot use Corollary 3.4, but the full criterion is still applicable. A computation gives
−¯ y1 y¯0 −¯ y3 y¯2 ψ= 2 y¯2 −¯ y1 y¯3 −¯ y0 y¯2 y¯12
( with kernel ρ =
y¯0 y¯2 y¯12 y¯1 y¯3 y¯1 y¯3 y¯0 y¯2 y¯22
)
The transpose of the first two vectors of ρ above are the “traditional” representatives of the inverse map Y → P1 . The ideal generated by the entries in these two vectors already generate the irrelevant ideal k[Y ]+ = (¯ y0 , y¯1 , y¯2 , y¯3 ), showing that the inverse is everywhere defined. Remark 3.8 A word about the defining equations of the image Y ⊂ Pm . As we have argued in several places, these equations are the “polynomial” equations among the defining equations of the Rees algebra of (f ), that is to say, they generate the subideal J(0,∗) ⊂ J of the presentation ideal of J . As such they are automatically found when one knows a minimal set of biforms generating the latter ideal. There is nevertheless both classical and modern interest in finding the defining equations of the image Y ⊂ Pm a priori. Some progress has been made recently in this problem in the case where the domain is the polynomial ring k[x] itself in the cases n ≤ 2. The methods vary in a range that encompasses Kronecker’s classical methods, homological methods and computational methods. It would be of some interest to have some of these methods extended to the case where k[X] still has low dimension but is no longer a polynomial ring. 10
Example 3.9 One classical illustration of the phenomenon in the last part of Remark 1.2 is the rational map F : X 99K P2 represented by the defining equations of a twisted cubic C ⊂ P3 , where X is a smooth general cubic surface containing C - the equation of X will be given by the determinant of a 3 × 3 matrix whose first two rows are the rows of the Hankel matrix defining C and the third row has entries which are sufficiently general klinear combinations of the coordinates of P3 . It is known that F is everywhere defined on X - geometrically, one argues that for every point p ∈ X there exists a twisted cubic not passing through p and whose defining equations still represent F . To see it via the main criterion, one simply computes Homk[X] (I, k[X]) as the kernel of the transposed matrix of the syzygy matrix φ of I on k[X] (here I ⊂ k[X] denotes the defining ideal of C read on k[X]). As one immediately checks, φ is the transpose of the original 3 × 3 matrix (up to k-linear combinations of the columns), hence is all linear (i.e., φ = φ1 in our standard notation) while Homk[X] (I, k[X]) is minimally generated by three vectors of k[X]-degree 2, one of which can be taken to be the transposed of the original representative of F . The transposed of any k-linear combination of the generators yield yet another representative of F , so these infinitely many choices retro-explain the geometric argument. Next, a computation gives that I is an ideal of linear type on k[X] - actually, this can also be verified theoretically since the symmetric algebra of I is Cohen–Macaulay and the relevant Fitting ideals of φ have the required rank (cf. [5]). Furthermore, since φ is linear, the matrix ψ is immediately guessed from φ. As it turns, its transpose ψ t is a 4 × 3 matrix in k[y0 , y1 , y2 ] of rank 3. Therefore, its cokernel is (isomorphic to) the ideal generated by its maximal minors. Applying Corollary 3.4, we conclude that F is birational onto P2 . As a bonus, we find the inverse map P2 99K X ⊂ P3 to be (uniquely) defined by cubics that generate a codimension two perfect ideal in k[y0 , y1 , y2 ] of algebraic multiplicity six, hence it is the homogeneous ideal of six general points in P2 . Example 3.10 An easy application of the criterion is to check that the Veronese map − 1, is birational onto its image (hence biregular). Of course, ν2 : Pn → PN , N = (n+1)(n+2) 2 this is not a big deal, however it shows a side curiosity, namely, that the syzygy matrix of the weak Jacobian dual matrix is precisely the (n + 1) × (n + 1) symmetric matrix whose 2 × 2 minors define the image of ν2 . In particular, the general representative of the inverse map is a linear combination of the rows of this matrix.
4
Cremona transformations
In this section we will consider rational maps F : Pn 99K Pn . Recall from the previous section that in the case of such maps, a representative f such that the ideal (f ) ⊂ k[x] has codimension at least two is uniquely defined and f is called the base ideal of F . A birational map F : Pn 99K Pn is called a Cremona transformation. We note that the main classical problem of these maps is to understand the structure of the group they form under composition of field maps. A major difficulty of this group is that it contains as subgroups the automorphisms groups of any of the affine pieces of Pn , while the latter are intractable as soon as n ≥ 3. One of the deepest theories in this direction is the so-called Mori theory. However, the tools in the this theory draw heavily 11
on methods of resolution of singularities and alike, making the transposition to ideal theory rather unaffordable. In any case, the goal of these notes is the study of an individual Cremona transformation, not the whole group.
4.1
Homological intrusion
For a finitely generated k[y]-module E, we denote by hd(E) its homologial dimension over k[y]. Note that E is torsionfree if and only if hd(EP ) ≤ ht (P ) − 1 for every prime P ∈ R, P ̸= 0, as follows from the classical Auslander–Buchsbaum formula in the local case. The translation of the main criterion in this case may be read as follows. Theorem 4.1 Let F : Pn 99K Pn stand for a rational map with base ideal f . The following are equivalent conditions: (i) F is a Cremona transformation (ii) f is algebraically independent over k and the weak Jacobian dual matrix ψ satisfies rankk[y] (ψ) = n and hd(coker(ψ)t )P ) ≤ ht (P ) − 1 for every prime P ∈ R, P ̸= 0. Moreover, when (ii) takes place, the inverse map is defined by the transpose of the unique generator of ker (ψ) (up to a nonzero factor from k).
4.2
Base ideal of linear type
The basic condition on an ideal to be used in this part is as follows. Definition 4.2 An ideal I ⊂ A in a ring A is said to be of linear type if the natural A-algebra homomorphism SA (I) RA (I) is injective. Proposition 4.3 (Linear Obstruction) Let F : Pn 99K Pn be a rational map whose base ideal I is of linear type. If F is a Cremona transformation then the matrix of linear syzygies of I have maximal rank. Remark 4.4 An important remark is that the converse is also true in characteristic zero and for monomial rational maps in arbitrary characteristic (see [12]). Moreover, in this case if the linear syzygy part generates a free submodule (necessarily of rank n) then the inverse map is defined by forms of degree n and the corresponding base ideal is a codimension 2 perfect ideal. This sort of behavior appears in a fragmentary way in classical examples but does not seem to have been isolated as a general pattern. Corollary 4.5 Let f ⊂ k[x] be n + 1 forms of the same degree such that dim k[x]/(f ) is a one-dimensional generically complete intersection at its minimal primes (equivalently, (f ) is a codimension n ideal which is generated by a regular sequence locally at its minimal primes). Then f defines a Cremona transformation of Pn if and only φ1 has maximal rank. In particular, this applies to the case where f = {f0 , f1 , f2 } ⊂ k[x0 , x1 , x2 ] define a plane rational map whose base ideal is generically a complete intersection on each of its base points. 12
Remark 4.6 In the case n = 2, the above local condition on the base ideal (f ) is also equivalent to requiring that the nth Fitting ideal I1 (φ) of (f ) have codimension three. This is the case if there are at least three distinct syzygy coordinates involving pure terms in x0 , x1 , x2 , respectively. Examples of such plane Cremona transformations are quadratic ones and, in general, the ones whose base points are reduced. They seem to fill a strategical sector of the plane Cremona group, certainly yielding the basic generators of the group (apart from collineations). The plane Cremona maps coming from homaloidal forms also fit there due to the result of [3].
4.3
The method of the bilinear algebra
For geometric purposes the linear type property is way too alien. If one is happy with a strictly classical geometric setup then linear presentation alone takes over. Thus, throughout this section we assume that the base field k is algebraically closed and of characteristic zero. Moreover we will only consider the case where the domain is Pn , but will allow the target to be any Pm . The basic result is a curious mix of algebra and geometry. Proposition 4.7 Let φ = (F0 : · · · : Fr ): Pr 99K Pr be a rational map where F0 , . . . , Fr are forms of the same degree generating an ideal I ⊂ k[x] of codimension ≥ 2. Set k[x, y] for the bihomogeneous coordinate ring of Pr × Pr and consider the bigraded incidence k-algebra A = k[x, y]/I1 (y · φ1 ), defined by the ideal of entries of the product matrix (y) · φ1 , where φ denotes a graded presentation matrix of I over k[x]. Finally, let R = Rk[x] (I) stand for the Rees algebra of the ideal I ⊂ k[x]. Then: (i) There is a surjective map of bigraded k-algebras ρ : A R; (ii) If the Jacobian determinant of F0 , . . . , Fr is nonzero and if ker (ρ) is a minimal prime of A then φ is a Cremona map ; (iii) If the Jacobian determinant of F0 , . . . , Fr is nonzero and if φ1 has maximal rank r then φ is a Cremona map. The following is an application of this method. One says that a homogeneous polynomial f ∈ k[x0 , . . . , xn ] is homaloidal if its partial derivatives define a Cremona transformation of Pn . Theorem 4.8 Let n ≥ 2 and let f denote the determinant of the following matrix (called sub–Hankel): x0 x1 x2 . . . xn−2 xn−1 x1 x2 x3 . . . xn−1 xn x2 x3 x4 . . . xn 0 .. .. .. .. .. . . . ... . . xn−2 xn−1 xn . . . 0 0 xn−1 xn 0 ... 0 0 13
Set J = (∂f /∂x0 , . . . , ∂f /∂xr ). Then: (i) For every value of i in the range 1 ≤ i ≤ n−1, the partial derivatives ∂f /∂x0 , . . . , ∂f /∂xi divided by their common g.c.d. define a Cremona transformation of Pi ; in addition, the base ideal of the inverse map is also a codimension two perfect ideal of linear type and both ideals are generated in degree i; (ii) The linear part of the graded presentation matrix of J has maximal rank; (r+1)(r−2)
(iii) The Hessian of f has the form h(f ) = c xr
, c ∈ k, c ̸= 0;
(iv) f is homaloidal.
References [1] W. Bruns and U. Vetter, Determinantal Rings, Lecture Notes in Mathematics, 1327, Springer-Verlag, Berlin-Heidelberg-New York, 1988. [2] B. Crauder and S. Katz, Cremona transformations with smooth irreducible fundamental locus, Amer. J. Math. 111 (1989), 289–309. [3] I. Dolgachev, Polar Cremona transformations, Mich. Math. J., 48 (2000), 191–202. [4] L. Ein and N. Shepherd-Barron, Some special Cremona transformations, Amer. J. Math. 111 (1989), 783–800. [5] J. Herzog, A. Simis and W. Vasconcelos, Koszul homology and blowing-up rings, in Commutative Algebra, Proceedings, Trento (S. Greco and G. Valla, Eds.). Lecture Notes in Pure and Applied Mathematics 84, Marcel-Dekker, 1983, 79–169. [6] K. Hulek, S. Katz, F.-O. Schreyer, Cremona transformations and syzygies, Math. Z. 209 (3) (1992), 419–443. [7] F. Russo and A. Simis, On birational maps and Jacobian matrices, Compositio Math. 126 (2001), 335–358. [8] J. G. Semple, On representations of the Sk of Sn and of the Grassmann manifolds G(k, n), Proc. London Math. Soc. 29 (1930), 200–221. [9] A. Simis, Remarkable graded algebras in algebraic geometry, XIII ELAM, IMCA, Lima, July 1999. [10] A. Simis, Cremona transformations and some related algebras, J. Algebra 280 (1) (2004), 162–179. [11] A. Simis, B. Ulrich and W. Vasconcelos, Jacobian dual fibrations, Amer. J. Math. 115 (1993), 47–75. [12] A. Simis and R. Villarreal, Linear syzygies and birational combinatorics, Results Math. 48 (2005), 326–343. 14
´ tica, CCEN, Universidade Federal de Pernambuco, Departamento de Matema ´ ria, 50740-540 Recife, PE, Brazil Cidade Universita E-mail:
[email protected]
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