Birational reparametrization of rational varieties of arbitrary dimension and rational surfaces of arbitrary codimension

Given a non-proper (unirational) rational parametrization of a variety that is known to be rational, the reparametrization problem asks to compute a proper (birational) parametrization together with a rational change of parameters, using only the parametric input. Caravantes, Pérez-Díaz and Sendra (arXiv:2211.07450; journal version 2024) solved this for rational surfaces in $\mathbb{P}^3$ by characterizing the solutions through the generic fiber of the input parametrization, and they explicitly posed its extension as future work: develop the theory of base points and generic fibers needed to compute birional reparametrizations of rational surfaces of arbitrary codimension and of rational varieties of arbitrary dimension, without invoking parametrization algorithms for implicitly defined varieties. No resolution of this extension problem is known to date. In dimension at least three the problem is additionally obstructed by the failure of unirationality to imply rationality, so rationality must enter as a hypothesis.

Background

Let $K$ be an algebraically closed field of characteristic zero. A rational parametrization of a projective variety $V\subset\mathbb{P}^n(K)$ is a tuple of rational functions defining a dominant rational map $\mathcal{P}:\mathbb{P}^r(K)\dashrightarrow V$. The parametrization is proper (equivalently, birational) when this map is generically injective. A measure of improperness is the degree of the map, $\mathrm{degMap}(\mathcal{P})$: the number of preimages of a general point of $V$, which equals the cardinality of the generic fiber $\mathcal{F}_g(\mathcal{P})$, the fiber taken over a point whose coordinates are independent transcendental elements. In computer-aided geometric design (CAGD) an improper parametrization traces every point of the variety $\mathrm{degMap}(\mathcal{P})$ times (its tracing index), which complicates rendering, trimming, intersection and meshing, so converting a merely unirational parametrization into a proper one is a basic algorithmic task.

The problem has two traditional entry points. From implicit equations one runs a parametrization algorithm: elementary for curves, and solved for surfaces over $K$ by Schicho (1998). From a parametric input one asks for a reparametrization: a birational parametrization $\mathcal{Q}$ of $V$ together with a rational map $\mathcal{S}:\mathbb{P}^r(K)\dashrightarrow\mathbb{P}^r(K)$ such that $\mathcal{P}=\mathcal{Q}\circ\mathcal{S}$. For curves, existence of a solution over any field is Luroth's theorem (every unirational curve is rational, and the rational function field of the curve embeds as a subfield of a one-variable rational function field), and effective reparametrization algorithms are classical (e.g. Pérez-Díaz 2006). For surfaces, testing properness, computing the degree of the map and inverting the parametrization were settled by Pérez-Díaz, Schicho and Sendra (2002) and Pérez-Díaz and Sendra (2004), but computing an actual reparametrization had only partial solutions reducing special cases to curves (Pérez-Díaz 2006, 2013), until Caravantes, Pérez-Díaz and Sendra (2022; journal version 2024) gave the first general algorithm for surfaces in $\mathbb{P}^3$.

Their method is fiber-based. The base locus of a parametrization is the set of parameter points at which all parametrizing homogeneous polynomials vanish simultaneously; the parametrization is called transversal when each base point is met by the moving curves of the underlying linear system with a suitable multiplicity condition, a technical hypothesis that lets the solution space be cut down. Theorem 1 of that paper characterizes the solution pairs $(\mathcal{Q},\mathcal{S})$ of $\mathcal{P}=\mathcal{Q}\circ\mathcal{S}$ in terms of the generic fiber of $\mathcal{P}$; Algorithm 2 computes such a pair in general, while under two additional hypotheses (the surface admits a birational parametrization with empty base locus, and $\mathcal{P}$ is transversal) the degree of the reparametrizing functions is known in advance and the solution space has smaller dimension (Algorithm 3). A declared design constraint is that the algorithms never invoke parametrization algorithms for implicitly defined surfaces: all steps (elimination, resultants, linear systems of plane curves) operate on the parametric input.

Everything in that paper is proved for rational projective surfaces in $\mathbb{P}^3$. Its Section 6 (Conclusions and future work) states the extension that defines this problem: adapt the strategy to compute birational reparametrizations of rational surfaces of arbitrary codimension ($V\subset\mathbb{P}^n$, $n\ge 4$, of dimension two) and of rational varieties of any dimension ($r\ge 3$), which first requires developing the theory of base points and generic fibers in those settings; the surface theory that would be generalized comprises the characterization of the solution pairs through the generic fiber (Theorem 1), the degree relations for the reparametrizing functions, and the base-point-based reductions of the solution space. The dimension-two ingredients that would need generalizing exist in pieces: the behavior of fibers and base points of parametrizations under projections (Pérez-Díaz and Sendra 2013) and formulas for the base point locus of surface parametrizations (Cox, Pérez-Díaz and Sendra 2022).

The extension is not routine, because the classical background changes abruptly beyond surfaces. Luroth's theorem for curves and Castelnuovo's rationality criterion for surfaces guarantee that a unirational variety of dimension at most two is rational; from dimension three on, unirational varieties need not be rational (smooth cubic threefolds are the classical counterexamples, and in 2025 Yang, Yu and Zhu even constructed nonrational varieties admitting unirational parametrizations of coprime degrees). A reparametrization can exist only if $V$ is rational, so in dimension at least three rationality must enter the problem as a hypothesis. In function-field terms, $\mathcal{P}$ induces an inclusion of the function field $K(V)$ into the purely transcendental extension $K(t_1,\dots,t_r)$, and computing a birational parametrization from $\mathcal{P}$ amounts to effectively realizing that subfield as purely transcendental: the effective Luroth problem, solved in dimensions one and two but open from dimension three on. Parametrizing rational varieties of dimension at least three from implicit equations is likewise open in general; the parametric, fiber-based strategy of the source paper is exactly the route around that obstacle which the proposed extension would test.

Problem Statement

Let $K$ be an algebraically closed field of characteristic zero and let $V\subset\mathbb{P}^n(K)$ be a rational projective variety, given together with a non-birational rational parametrization $\mathcal{P}:\mathbb{P}^r(K)\dashrightarrow V$ (so $\mathrm{degMap}(\mathcal{P})\ge 2$), in either of the two classes not covered by the existing theory:

(1) $r=2$ and $n\ge 4$: a rational surface embedded in projective space of codimension at least two; or

(2) $r\ge 3$: a rational variety of dimension at least three, with the rationality of $V$ assumed as a hypothesis rather than certified by a birational parametrization supplied with the input (unirationality alone does not imply rationality in these dimensions).

Develop the theory of the generic fiber and of the base locus of such parametrizations, and use it to design and prove correct a terminating algorithm which, without invoking parametrization algorithms for implicitly defined varieties, computes a birational parametrization $\mathcal{Q}:\mathbb{P}^r(K)\dashrightarrow V$ of $V$ together with a rational map $\mathcal{S}:\mathbb{P}^r(K)\dashrightarrow\mathbb{P}^r(K)$ such that

$\mathcal{P}(t)=\mathcal{Q}(\mathcal{S}(t)).$

The general-setting theory of base points and generic fibers sought here parallels the surface theory of Caravantes-Pérez-Díaz-Sendra, which comprises a characterization of the solution pairs $(\mathcal{Q},\mathcal{S})$ in terms of the generic fiber of $\mathcal{P}$ (their Theorem 1), degree relations for the reparametrizing functions, and base-point-based reductions of the solution space. The target comprises both classes above.

References

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