Computing a finite set of rational parametrizations covering a projective rational surface with arbitrary base locus

Alcázar and Pérez-Díaz (Adv. Appl. Math. 123 (2021) 102128) compute the form of highest degree of the implicit equation of a rationally parametrized surface - the part that cuts out the curve at infinity - but their method misses components of that curve whenever the parametrization has base points that blow up to curves at infinity, and they state that no algorithm is known to reparametrize this defect away and that no general algorithm computes a finite family of rational parametrizations covering a projective rational surface. Caravantes, Sendra, Sevilla and Villarino (Mathematics 9(4):338, 2021) subsequently settled the covering question for parametrizations with simple base points: three rational maps suffice and are computable. The problem is the remaining general case: a terminating algorithm that, from an arbitrary rational parametrization of a projective rational surface with unrestricted base locus (non-simple and infinitely near base points included), computes finitely many rational parametrizations whose images cover the entire surface - or, alternatively, a single reparametrization with no base point blowing up to a curve at infinity. No published work resolves the unrestricted case for singular surfaces or by an algorithm driven by the given parametrization.

Background

A rational surface is an algebraic surface admitting a birational map from the projective plane $\mathbb{P}^2$; in geometric modeling it is usually given by a rational parametrization $\mathcal{P}(\bar{t})=(p_{11}(\bar{t})/p(\bar{t}),\,p_{21}(\bar{t})/p(\bar{t}),\,p_{31}(\bar{t})/p(\bar{t}))$ with polynomial entries over an algebraically closed field $\mathbb{K}$ of characteristic zero. Homogenizing the parameters $(t_1,t_2)$ with a third variable gives a projective parametrization $\mathcal{P}^*_h(T)=(p^h_{11}(T):p^h_{12}(T):p^h_{13}(T):p^h(T))$, a rational map $\mathbb{P}^2\dashrightarrow\mathcal{V}^*\subset\mathbb{P}^3$ onto the projective closure $\mathcal{V}^*$ of the surface. The common zeros of the homogeneous coordinate forms are the base points of the parametrization; the parametrization is called proper (birational) when a generic point of the surface has exactly one preimage, and practical parametrizations need not be proper.

A rational map is undefined at its base points. The standard resolution is to blow up each base point - that is, to replace the point by the set of one-dimensional directions through it, an exceptional curve isomorphic to $\mathbb{P}^1$ - which turns the parametrization into a morphism from the blown-up plane onto $\mathcal{V}^*$. The image of each exceptional curve is either a point or a curve of $\mathcal{V}^*$, and the points of such an image curve are reached by the resolution but by no point of the original parameter plane: the parametrization misses them. The critical situation is when missed curves lie in the hyperplane at infinity $\{x_4=0\}$ - one then says the base points blow up to a curve at infinity - because the curve at infinity $\mathcal{V}^*\cap\{x_4=0\}$ is cut out by the form of highest degree $f_d(\bar{x})$ (the leading homogeneous form) of the implicit equation $f(\bar{x})=0$ of the surface, and components missed by the parametrization cannot be recovered from it. The leading form governs the asymptotic behavior of the surface and enters surface recognition (surfaces of revolution, translational surfaces, cyclides) and symmetry detection, which is why one wants $f_d$ without paying for the full implicit equation.

Alcázar and Pérez-Díaz (2021) compute $f_d$ directly from the parametrization: each irreducible factor $q_i$ of the denominator $p$ contributes the implicit equation of the image of the planar rational map that the parametrization induces on the plane curve $q_i=0$, raised to an exponent built from the multiplicity of $q_i$ in $p$ and from the degrees - the cardinalities of generic fibers - of that induced planar map and of the surface parametrization itself; a comparison of these degrees detects whether components at infinity were missed. The method succeeds exactly when no base point blows up to a curve at infinity. As remedies for the bad case the authors propose reparametrizing the surface so the defect disappears, or covering the projective surface with several parametrizations; for the surface $x_3-x_1x_2=0$, the parametrization $(t_1,t_2,t_1t_2)$ has two base points each blowing up to a line at infinity, while the reparametrization $(1/t_1,1/t_2,1/(t_1t_2))$ has base points that do not. They state that no algorithm is known to compute such a reparametrization in general, and that no general algorithm computes a finite set of rational parametrizations covering a projective rational surface.

For the covering question the partial results at the time were: every affine rational surface without projective base points is covered by at most three rational parametrizations (Sendra-Sevilla-Villarino, ISSAC 2014), and affine ruled surfaces are covered by two, computably (Sendra-Sevilla-Villarino, 2017). Caravantes-Sendra-Sevilla-Villarino (2021) then settled the projective case under a hypothesis on the input: if the base locus is empty or the Jacobian matrix of the coordinate forms has rank two at every base point (the base points are simple; multiple base points, and infinitely near base points - base points appearing on exceptional curves after a first blow-up - are excluded), an algorithm outputs three rational maps from the affine plane whose images cover the projective surface, and two maps cover the affine part; they explicitly leave open the case where no condition is imposed on the base locus, and how many parametrizations are really necessary in the projective case. In the smooth case the covering question has a geometric answer: every smooth rational projective surface over $\mathbb{C}$ is the union of three Zariski open subsets isomorphic to the affine plane (Caravantes-Sendra-Sevilla-Villarino, RACSAM 117(2):69, 2023), and the three inclusions are rational parametrizations covering it; this construction proceeds from the surface itself rather than from an arbitrary input parametrization, so it does not settle the present problem. That several parametrizations are genuinely necessary is known: some affine rational surfaces admit no birational surjective parametrization from the plane (Caravantes-Sendra-Sevilla-Villarino 2018), and some admit no finite surjective parametrization (Ballico-Fontanari 2022).

Problem Statement

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. The input is a projective rational parametrization $\mathcal{P}^*:\mathbb{P}^2_{\mathbb{K}}\dashrightarrow\mathcal{S}\subset\mathbb{P}^n_{\mathbb{K}}$, given by $n+1$ coprime homogeneous polynomials of equal degree, whose image is Zariski dense in the rational projective surface $\mathcal{S}$ (that is, not contained in any proper algebraic subset of $\mathcal{S}$); $\mathcal{P}^*$ may be improper and its base locus is unrestricted - base points at which the Jacobian of the coordinate forms has rank $<2$ (non-simple, multiple base points) and infinitely near base points arising along blow-ups are all allowed. Let $H_\infty$ denote the designated hyperplane at infinity of the embedding $\mathcal{S}\subset\mathbb{P}^n$; the curve at infinity is $\mathcal{S}\cap H_\infty$.

The problem, stated as open by Alcázar-Pérez-Díaz (2021) and still open in this generality, is to compute the curve at infinity $\mathcal{S}\cap H_\infty$ (equivalently, for $\mathcal{S}\subset\mathbb{P}^3$, the form of highest degree of its implicit equation) without full implicitization; each of the two open directions below yields this curve:

(1) Covering form. Design a terminating general algorithm that outputs finitely many explicitly given rational parametrizations $\mathcal{P}_1,\dots,\mathcal{P}_k:\mathbb{A}^2\dashrightarrow\mathcal{S}$ such that every point of $\mathcal{S}$ - in particular every point of the curve at infinity $\mathcal{S}\cap H_\infty$ - is the image of some parameter point under some $\mathcal{P}_i$, together with a proof of termination and of this covering guarantee for all inputs.

(2) Reparametrization form. Design a terminating general algorithm that outputs a single rational parametrization of $\mathcal{S}$ none of whose base points blows up to a curve at infinity - equivalently, whose image already contains the entire curve at infinity - again with proof.

Partial answers restricted to simple base points (Jacobian of rank two at each base point) already exist (Caravantes-Sendra-Sevilla-Villarino 2021) and do not by themselves resolve this problem.

References

1. Juan Gerardo Alcázar, Sonia Pérez-Díaz: Computing the form of highest degree of the implicit equation of a rational surface. Advances in Applied Mathematics 123 (2021), article 102128. DOI 10.1016/j.aam.2020.102128, https://doi.org/10.1016/j.aam.2020.102128.
2. Jorge Caravantes, J. Rafael Sendra, David Sevilla, Carlos Villarino: Covering Rational Surfaces with Rational Parametrization Images. Mathematics 9(4) (2021), article 338. DOI 10.3390/math9040338, https://doi.org/10.3390/math9040338 (arXiv:2101.07011, https://arxiv.org/abs/2101.07011).
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