Covering rational surfaces with rational parametrization images

Let $S$ be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps $f,g,h:\mathbb{A}^2 --\to S\subset \mathbb{P}^n$ such that the union of the three images covers $S$. As a consequence, we present a second algorithm that generates two rational maps $f,\tilde{g}:\mathbb{A}^2 --\to S$, such that the union of their images covers the affine surface $S\cap \mathbb{A}^n$. In the affine case, the number of rational maps involved in the cover is in general optimal.


Introduction
Rational parametrizations of algebraic varieties are an important tool in many geometric applications like those in computer aided design (see e.g. [Farin et al., 2002], [Hoschek and Lasser, 1993]) or computer vision (see e.g. [Marsh, 2005]). Nevertheless, the applicability of this tool can be negatively affected if the parametrization is missing basic properties: for instance its injectivity, its surjectivity, or the nature of the ground field where the coefficients belong to; see e.g. the introductions of the papers [Sendra et al., 2015], [Sendra et al., 2014] and [Sendra et al., 2017] for some illustrating examples of this phenomenon.
In this paper we focus on the surjectivity of rational parametrizations. Surjective parametrizations, also called normal parametrizations, have been studied by many authors but still many important questions, both theoretical and computational, stay open. The case of curves, over algebraically closed fields of characteristic zero, is understood comprehensively, and one can always find a surjective, indeed also injective, affine parametrization of the affine algebraic curve (see e.g. [Andradas and Recio, 2007], [Rubio et al., 2006], [Sendra, 2002], [Sendra et al., 2008]). Furthermore, the case of curves defined over the field of the real numbers is also studied and characterized in [Sendra, 2002].
The situation, as usually happens, turns to be much more complicated when dealing with surfaces. In [Bajaj and Royappa, 1995], [Gao and Chou, 1991] the first steps in this direction were given and answers for certain types of surfaces, like quadrics, were provided. Also, in [Pérez-Díaz et al., 2010] the relation of the polynomiality of parametrizations to the surjectivity was analyzed. Nevertheless, in [Caravantes et al., 2018] it is shown that there exist rational surfaces that cannot be parametrized birationally and surjectively. As a consequence of this fact, the question of whether every rational surface can be covered by the union of the images of finitely many birational parametrizations is of interest. The answer is positive and can be deduced from the results in [Bodnár et al., 2008]. In previous papers, the authors have studied this problem for special types of surfaces. In [Sendra et al., 2015] the unreachable points of parametrizations of surfaces of revolution are characterized. In [Sendra et al., 2017] it is proved that ruled surfaces can be covered by using two rational parametrizations. In addition, in [Sendra et al., 2014] an algorithm to cover an affine rational surface without based points at infinity with at most three parametrizations is presented.
In this paper we continue the research described above and we present two main algorithmic and theoretical results. Moreover, we provide an algorithm that, for any projective surface parametrization, generates a cover of the projective surface with three parametrizations, assuming that, either the base locus of the input is empty, or the Jacobian of the input parametrization, specialized at each base point, has rank two. As a consequence of this result, we also present an algorithm that, for a given affine parametrization whose projectivization satisfies the condition on the base points mentioned above, returns a cover of the affine surface with two affine parametrizations.
Taking into account the results in [Caravantes et al., 2018], the affine cover presented in this paper is, in general, optimal. Furthermore, it improves the results in [Sendra et al., 2014] and extends the results in [Sendra et al., 2017] to a much more general class of surfaces. With respect to the projective cover case, although theoretically interesting, we cannot ensure that the number of parametrizations involved in the cover is optimal for a generically large class of projective surfaces since, for instance, the whole projective plane can be covered with just two maps from the affine plane as the following example shows. We leave this theoretical question open as future reseach Example 1.1. Consider the following two maps: Then for any point P := (x : y : z) ∈ P 2 , if x = 0 then P := f y x , z x ; if x = 0 = z then P := g 0, y z ; and if x = z = 0 then P := g(1, y), y = 0. This means that we can cover the whole projective plane with just two maps from the affine plane.
The paper is structured as follows. In Section 2, we introduce some notation and we briefly recall some notions and results that will be used throughout the paper. In Section 3, we present the projective cover algorithm and in Section 4 we illustrate the result by means of some examples. In Section 5, we apply the results in Section 3 to derive the two affine parametrization cover algorithm.

Preliminaries
In this section, we briefly recall some concepts and results that will be used in the subsequent sections. We essentially recall some results on the fundamental locus of rational maps and some consequences and the characterization of zero dimensional ideals via Gröbner bases. Throughout this paper K is an algebraically closed field of characteristic zero, and P n the projective space over K. Moreover, we denote by A i the affine space {(x 0 : · · · : x n ) ∈ P n , | x i = 0}. In the examples, the field K will be the field C of the complex numbers.
Let X be an irreducible projective variety and let f : X P n be a rational map. The fundamental locus of f is the algebraic set F (f ) of points to which f cannot be extended regularly. Any P ∈ F (f ) is called a fundamental point of f . The following theorem analyzes the dimension of the fundamental locus.
Theorem 2.1. [Hartshorne, 1977, Lemma V.5.1] Let X be a smooth irreducible projective variety and let f : X P n be a rational map generically finite. The fundamental locus of f has codimension at least 2 in X.
Corollary 2.2. Let X be a smooth irreducible surface and f as in Theorem 2.1. F (f ) is either empty or zero dimensional.
The traditional way for solving indeterminacies in algebraic geometry consists in blowing up fundamental points (see e.g. [Shafarevich, 1994, IV.3.3]) and composing with the corresponding map as the next theorem shows.
Theorem 2.3. [Hartshorne, 1977, Example 7.17.3] or [Beauville, 1983, Theorem II.7] Let X be a smooth surface. Let f : X P n be a rational map. Then there exists a commutative diagram where g is a composite of blowups involving fundamental points of f and h is a morphism.
A first consequence of Theorem 2.3 is the following.
Corollary 2.4. [Caravantes et al., 2018, Corollary 2.5] Let X and f be as in Theorem 2.3. For any fundamental point P of f , h(g −1 (P )) is a connected finite union of rational curves.
Remark 2.5. Let f : P 2 P n be as in Theorem 2.3, and let S be the Zariski closure of f (P 2 ) in P n .
Such a subset consists of some rational curves and, if f contracts L ∞ , a closed point (see [Caravantes et al., 2018, Corollary 2.5

]).
We end this section with a well-known result on elimination theory that will be used in Section 3.
Theorem 2.6. [Cox et al., 2015, Chapter 5, Theorem 6] Let I be an ideal in K[x 1 , .., x n ]. Then, the following statements are equivalent: 1. The algebraic subset of K n defined by I is a finite set.
2. Let B be a Gröbner basis for I with respect to a fixed monomial ordering. Then, for each 1 ≤ i ≤ n, there is some m i ∈ N such that x mi i is the leading monomial of an element of B.

Covering projective surfaces with three parametrizations
Throughout this section, let S ⊂ P n be a rational projective surface and let F = (F 0 : · · · : F n ) : P 2 S ⊂ P n be a (not necessarily birational) parametrization of S, given by n+1 homogeneous coprime polynomials F 0 , ..., F n where the nonzero polynomials have degree d. In addition, let the homogeneous ideal I = (F 0 , ..., F n )K[x 0 , x 1 , x 2 ] be called the fundamental ideal associated to F .
Since the polynomials defining F are coprime, by Corollary 2.2, I defines a closed algebraic subset A of P 2 that is either empty or consists of a finite amount of points. If A = ∅, then F defines a regular map and its restrictions to each of the three affine planes A i = {(x 0 : x 1 : x 2 ) | x i = 0} covering P 2 define 3 charts that cover S, since the image of a projective variety by a regular map is always Zariski closed. Otherwise, say A = {P 1 , ..., P k }; we need to make the following assumption: ( * ) If A = ∅, then for every P ∈ A := {P 1 , . . . , P k } the jacobian matrix of F at P has rank 2.
We observe that the real constraint lays in ( * ), since conditions (a) and (b) are satisfied after a general change of coordinates. In the following remark we discuss how these hypotheses can be computationally checked.
1. Note that condition ( * ) implies that the point P j is regular in the projective scheme defined by the ideal I. Then, the intersection multiplicity is 1 at every P j . Now, if we consider the ideal J defined by the 3 × 3 minors of the jacobian matrix of F , the following methods, among others, can be applied to test ( * ): e. the irrelevant ideal) or, equivalently, whether I + J contains a power of any of the variables.
2. Checking (i) without explicitly determining P 1 , ..., P k can be carried out by certifying that all the ideals generated by {F 0 , ..., F n , x i } (i = 0, 1, 2) either are zero-dimensional or contain a power of the irrelevant ideal.
3. Checking (ii) can be done by computing the ideal bases B 1 and B 2 of Algorithm 3PatchSurface and checking whether they have adequate shape (see Proposition 3.3). However, it may be more efficient to check that, for all i = 0, 1, 2, the gcd of all resultants of couples (F 0 , F j ) with respect to x i is square free.
Now we consider the three affine planes A i defined above. According to (a) all P j lie in the intersection of the three affine planes. In this situation, the strategy is as follows. We will work with the parametrization f := F | A0 : A 0 S as defined in A 0 , and we will blowup A 1 and A 2 at the base points of F to get new affine planes A 1 and A 2 with projections Bl 1 : A 1 → A 1 and Bl 2 : A 2 → A 2 . Now, we introduce the compositions and we prove that the union of the images of f , g and h is the whole S (see details in Proposition 3.3). During this process, we also need to keep track of what happens with the infinity line of A 0 , namely As a consequence, we derive the following Algorithm 3PatchSurface.
Remark 3.2. In Proposition 3.3 we show that the integer k, introduced in the algorithm, is exactly the number of base points, that is the cardinality of A (see above), so we have not introduced equivocal notation.

Algorithm 3PatchSurface
Require: A map F = (F 0 : · · · : F n ) defined by coprime homogeneous polynomials in K[x 0 , x 1 , x 2 ], where the nonzero polynomials have the same degree, parametrizing a Zariski dense subset of a projective surface S ⊂ P n , such that conditions ( * ), (a) and (b) are satisfied. Ensure: Two maps G = (G 0 : · · · : G n ) and H = (H 0 : · · · : H n ) defined by homogeneous polynomials in K[x 0 , x 1 , x 2 ] where the nonzero polynomials have the same degree (while in the same list), such that the union of the images of F (1 : : ), G( : 1 : ), H( : : 1) : In the following, we see that the output of Algorithm 3PatchSurface is correct (see Theorem 3.5). We also recall that the required conditions for the algorithm can be checked computationally according to Remark 3.1. We start by proving that Step 4 works properly, assuming that conditions ( * ), (a) and (b) are satisfied. This is probably a well-known result in a more general setting but, up to the authors' knowledge, there are no suitable references for the proof.
Proposition 3.3. Let F = (F 0 : · · · : F n ), I 1 and I 2 be as in Algorithm 3PatchSurface. There exist p 1 , q 1 ∈ K[x 2 ], p 2 , q 2 ∈ K[x 1 ] such that k =deg(p i ) > deg(q i ) for all i = 1, 2, and the reduced Gröbner basis B 1 and B 2 of I 1 and I 2 respectively, have the following shape: Proof. Observe that both I 1 and I 2 define finite sets in A 1 and A 2 , respectively. By Theorem 2.6, this implies that, since B 1 and B 2 are Gröbner bases, with respect lex(x 0 < x 1 < x 2 ), there is a polynomial in each B i just involving x 0 . This is p 1 for I 1 and p 2 for I 2 . Due to ( * ), (a) and (b), each p i defines k different parallel lines in A i , so its degree is k.
Applying again Theorem 2.6, there is another monic polynomial in K[x 0 ][x 1 ] ∩ I 1 . Since the basis is reduced, and x 1 − 1 was originally among the generators of I 1 , this polynomial in K[x 0 ][x 1 ] for I 1 is precisely x 1 − 1. Now, let q i be the interpolation polynomial whose graph goes through all the base points P j ∈ A i of F . It does exist because (b) holds (so there are no two different P j in the same vertical line) and its degree is at most k − 1. Then x 1 − q 2 (x 0 ) vanishes at every P j so, by Hilbert's Nulltellensatz, it belongs to √ We apply an analogous argument and reduction by x 1 − 1 to deduce that the monic polynomial in K[x 0 , x 1 ][x 2 ] for I 1 must be x 2 − q 2 (x 0 ), and we know that x 2 − 1 is a reduced monic polynomial for I 2 , so it is in B 2 .
Since B 1 and B 2 are reduced Gröbner bases with respect to the lexicographical order x 0 < x 1 < x 2 and the ideals they generate define, precisely, the fundamental locus, they are the reduced Gröbner bases of I 1 and I 2 .
Before continuing, we state a Lemma.
Proof. By the properties of the rational map F : P 2 P n , defined by F , and the fact that a blow up is bijective outside its blown up points, we know that any base point of G( : 1 : ) = F • Bl 1 ( : 1 : ) would be in one of the lines Bl 1 (P j ). Now, we fix P j = (1 : α j : β j ) and we then prove that G( : 1 : ) has no base points in the line {x 0 = 1 αj } ⊂ A 1 . Since all F i (x 0 : 1 : x 2 ) vanish at P j , they have α j x 0 − 1 as a common factor (note that α j = 0 because (a) holds). Then, since

This means that
On the other side, the vector (1, α j , β j ) is also in the kernel of the jacobian matrix of F at P j , due to Euler's formula for homogeneous polynomials: , the Jacobian matrix of F has rank 2 at P j , so (1, α j , β j ) generates its kernel. Then, 1, 0, ∂q1 ∂x0 1 αj + ∂p1 ∂x0 1 αj x 2 is not in such kernel, since α j = 0. Therefore, for all x 2 ∈ C. Since all entries of G( 1 αj : 1 : x 2 ) are divisors of those of G( 1 αj : 1 : x 2 ), then G is nonzero throughout the whole line x 0 = 1 αj in A 1 .
Repeating the argument with H finishes the proof.
The next result states the correctness of Algorithm 3PatchSurface: Theorem 3.5. The three parametrizations F , G and H output by Algorithm 3PatchSurface satisfy that the union of the images of F (1 : : ), G( : 1 : ) and H( : : 1) covers S completely.
• ϕ has no base points.
In this situation, we observe that, since ϕ has no base points, then it is a regular morphism, so ϕ(Bl P1,...,P k (P 2 )) is an algebraic subset of P n . On the other side, F is dominant over S, so and, hence, the theorem holds.
First of all, we define j 1 and j 2 . The blow up of all the base points of F is, locally, the blow up of the ideal I i in A i with i being 1 or 2. Knowing the bases of I 1 and I 2 , we have (see [Shafarevich, 1994, Section 4.2]): While the way to glue the two open subsets is not interesting for the purpose of this proof, it is easy to see that Bl I1 (A 1 ) ∪ Bl I2 (A 2 ) is the whole Bl {P1,...,P k } (P 2 ) minus the point that is the strict transform of (1 : 0 : 0). We now consider the inclusions x 1 : 1)) .
On the other hand, A i covers the whole A i except the vertical lines through the base points. These affine lines are completely contained in A 0 , since the infinity point is (0 : 0 : 1) for A 1 and (0 : 1 : 0) for A 2 . This means that, to show that A 1 , A 2 and A 0 \{P 1 , ..., P k } -through j 1 , j 2 and the blowup j 0 of the base points-cover Bl {P1,...,P k } (P 2 ), we just need to prove that A 1 and A 2 cover the exceptional divisor. So we fix P j = (1 : α j : β j ) and we call E j ≃ P 1 the component of the exceptional divisor corresponding to P j . Note that Bl i covers a full neighborhood of P j in A i minus the vertical line. This corresponds to the line joining P j with (0 : 0 : 1) for the case i = 1 and the line joining P j with (0 : 1 : 0) for the case i = 2. By condition (b), these lines do not contain other base points. These two lines represent two different directions at P j (i.e. two different points in E j ). This means that j 1 ({x 0 = 1 αj }) covers all E j except one point and that j 2 ({x 0 = 1 βj }) is an affine line passing through that point.
The only task remaining is proving that ϕ has no base points. Such base points would be in the exceptional divisor, which is covered by A 1 and A 2 , but Lemma 3.4 states that there are no base points of ϕ in j 1 ( A 1 ) ∪ j 2 ( A 2 ).

Examples
This section is devoted to illustrating Algorithm 3PatchSurface by examples. We start with a toy example in which we explicitly show that the three parametrizations cover the whole projective surface.
3. if y 2 = 0 and y 0 = 0, then, taking y 0 = 1, Note here that every A in this case is gotten twice. Even Observe that F (1 : : ) covers the whole umbrella minus a couple of rational curves. Then, G( : 1 : ) covers these curves minus just a point, that is covered by H( : : 1).
The following example applies Algorithm 3PatchSurface to a classic surface. While the computation time is not long, the output is too large, so we just sketch some computations.

The affine case
In this section, we slightly change our point of view and we consider the problem of covering a rational affine surface by means of the images of several affine parametrizations. So, in the sequel we consider that we are given F and S as in Section 5, and we deal with the problem of covering S ∩A n , where A n is the open subset of P n defined by the first variable not vanishing. Equivalently, one may consider that we are indeed given an affine parametrization and the affine surface that it defines. Nevertheless, to be consistent with the notation used throughout the paper, we will use the first notational statement of the problem.
In this section, we prove that to cover a rational affine surface, only two patches are necessary (see Theorem 5.3 and Corollary 5.6). The basic idea is as follows. The given parametrization F (1 : x 1 : x 2 ) covers a constructible subset. The complement of such subset is contained in the image of L ∞ (that is, F (0 : x 1 : x 2 )) and the base points, which is a finite union of affine rational curves (see Corollary 2.4) and, maybe, an isolated point corresponding to a contracted L ∞ . The parametrization G(x 0 : 1 : x 2 ) of Algorithm 3PatchSurface, restricted to certain vertical lines, covers all such affine curves except at most one point. Since such curves also have a point at infinity, we want such point to be the image of the Algorithm 2PatchForAffine Require: A list F = (F 0 : · · · : F n ) of coprime homogeneous polynomials of the same degree in K[x 0 , x 1 , x 2 ], parametrizing a Zariski dense subset of a projective surface S ⊂ P n C , such that conditions ( * ), (a) and (b) are satisfied. Ensure: A list g ′ = (g ′ 0 : · · · : g ′ n ) of rational functions of two variables such that F (1 : : ) and g ′ ( , ) cover S ∩ A n . 1: Compute B 1 = {x 2 − q 1 (x 0 ), x 1 − 1, p 1 (x 0 )} and G = (G 0 : · · · : G n ) as in Algorithm 3PatchSurface. 2: for α root of p 1 (x 0 ) or α = 0 do 3: if deg G 0 (α : 1 : x 2 ) < max{deg G i (α : 1 : x 2 ) | i = 1, ..., n} or G 0 (α : 1 : x 2 ) is constant then 4: Include α in set A and β α := ∞. Include α in set B. Choose β α among the roots of G 0 (α : 1 : x 2 ).

7:
end if 8: end for 9: Let s(x 0 ) be a polynomial vanishing at all the α ∈ A and not vanishing at any of the α ∈ B. See Remark 5.1 below, for suggestions on how to find one. 10: Choose r(x 0 ), a polynomial such that r(α) = β α s(α) for all α ∈ B and r(α) = 0 for all α ∈ A. 11: Find Bezout coefficients u and v such that gcd(r, s) = u · r + v · s.
point at infinity of the parameter line. Based on these ideas, we derive Algorithm 2PatchForAffine that, when the original parametrization satisfies ( * ), (a) and (b), covers affine surfaces using just two parametrizations.
Remark 5.1. Let us comment some computational aspects of Algorithm 2PatchForAffine.
1. For s(x 0 ), one may proceed as follows: collect the coefficients of G 0 (x 0 : 1 : x 2 ) for x 2 except the one for x 0 2 , and compute the gcd, d(x 0 ), of these coefficients; then s(x 0 ) = gcd(d(x 0 ), x 0 p 1 (x 0 )). 2. Note that Algorithm 3PatchSurface does not extend the field that is used to define F , However, Algorithm 2PatchForAffine needs to consider possibly algebraic coordinates for the points that need to be sent to the infinity in the parameter plane A 1 . Example 5.4 shows that the application of the algorithm forces the usage of algebraic coefficients.
In order to state the correctness of Algorithm 2PatchForAffine, we start with a technical lemma.
Let us, now, prove that Algorithm 2PatchForAffine works as expected.
Taking into account how G and H in Algorithm 3PatchSurface are defined, any point in A 1 and A 2 , not in L ∞ or in {p i (x 0 ) = 0}, is sent by G(x 0 : 1 : x 2 ) or H(x 0 : x 1 : 1) into the image by F of a point in A 0 . Therefore, we need to check that g ′ covers any affine point in G is either a point or a rational curve covered by G(α : 1 : x 2 ) ∪ H(α ′ : x 1 : 1), where either α = α ′ = 0 or (1 : 1 α : 1 α ′ ) is a base point of F . Moreover, such component is the Zariski closure of the image of the restriction G| {x0=α,x1 =0} , which coincides with the Zariski closure of g ′ | {x0=α} .
If C is just a point, then it is covered by g ′ | {x0=α} . Otherwise, it is well known that any morphism defined in an open subset of a projective smooth curve can be extended regularly to the whole curve and that the image of a projective curve by a regular morphism is a Zariski closed subset. Then, we can extend g ′ | {x0=α} to the Zariski closure of the line where it is defined and we would cover completely C. This means that g ′ | {x0=α} covers all C minus, at most, just a point (the image of the infinity point of the affine line). However, this point is G(α : 1 : β α ), which is at infinity by Lemma 5.2. Therefore, any point in C ∩ {y 0 = 0} is in g ′ ({x 0 = α}) ⊂ g ′ (A 2 ).
Example 5.4. Consider the following projective transformation of the Veronese morphism: Since there are no base points, Algorithm 2PatchForAffine generates G = F , and then, for just α = 0, it computes β 0 such that β 2 0 + 1 = 0. This means that β 0 must be imaginary, so it is not rational. The output has imaginary coefficients.
We observe that any choice of two rationally defined affine planes A 0 and A 1 of the projective plane will leave a point P in the projective plane over Q out of the union, and then F (P ), which is not at infinity, will not be covered. Observe, however, that the surface is isomorphic to the projective plane, so one can compose the two maps appearing in Example 1.1 with the Veronese morphism to cover, not just the affine part, but the whole projective surface without extending the field.
Example 5.5. Let us again consider the cubic of Example 4.2. We recall that One can factor G 0 , as obtained in Example 4.2, and one of the factors is Here, one can easily get x 2 as a rational function on x 0 , so we have a point going to infinity at each vertical line: The set A is given by the common roots of p 1 and the denominator in (1), so We need a polynomial r(x 0 ), coprime with s(x 0 ), whose values at the roots of x0p1 (x0) s (x0) equal β x0 s(x 0 ).
In [Caravantes et al., 2018], it is proved that there exist affine surfaces that cannot be covered by means of a unique map from the affine plane. In fact, the surface in Example 5.5 is proved to be one of them. Now, the following corollary of Theorem 5.3 shows that, under hypotheses (*), (a), (b), one can always cover the affine surface with two affine parametrization images.
Corollary 5.6. Let S be an affine surface such that there exists a parametrization f : A 2 S with a projectivization F satisfying ( * ), (a) and (b). Then S can be covered with just two parametrizations.
In order to prove that two affine patches are enough we have had to impose, to the projectivization of the input affine parametrization, hypotheses (*), (a) and (b). If we do not impose (*), we cannot ensure this general result. However, it is interesting to observe that there are affine surfaces not satisfying ( * ) that can be covered by only one map. To create an example, it is enough to send the exceptional divisor to the infinity hyperplane together with the image of L ∞ .

Conclusions
Covering a rational surface as the union of images of several parametrizations is an important problem with direct implications in the potential feasibility of the applications. There were previous results for some particular types of surfaces. In this paper we enlarge the class of surfaces where this approach is valid. More precisely, we present two algorithms, one for the projective case and another for the affine case. In the latter, we are able to cover any rational affine surface, satisfying certain hypotheses on the base locus of the input parametrization, that is optimal in the number of cover elements, namely, two. For the projective case, the answer provides three cover parametrizations. Some open problems are, on one hand, the extension of the results to the case where no condition of the base locus is imposed, and the optimality on the number of cover parametrizations in the projective case.