Effective base-point data of rational surface parametrizations from resultants: genericity tests, individual multiplicities, and the genus of the auxiliary curves

Cox, Pérez-Díaz and Sendra (arXiv:2008.08009; Communications in Mathematics and Statistics 10(4) (2022) 757-783) proved that the total multiplicity of the base point locus of a projective rational surface parametrization equals the degree of the content of an explicit univariate resultant, computed after a generic projective change $L$ of the target coordinates, and used it to give a new proof of the degree formula relating parametrization degree, surface degree, base point multiplicity and map degree. The method is effective only generically, and the paper's proof leaves three linked effectiveness gaps; of these, only the third is posed explicitly as a question in the paper, while the first two are effectiveness gaps in the paper's text. First, the identities hold on non-empty Zariski open subsets of the transformation group ($\Omega$, $\Omega'$, $\Omega_{\mathcal{S}}$, $\Omega'_{\mathcal{S}}$, $\Omega''_{\mathcal{S}}$), and no packaged algorithm decides for a concrete $L$ whether the formulas apply. Second, the resultant content yields only the sum of the pointwise Hilbert-Samuel multiplicities $e(I_A,R_A)$ of the base points; no elimination-theoretic procedure extracts the individual values. Third, the auxiliary curves $\mathscr{C}(K_i^L)$ governing the elimination can have genus 1 even for birational parametrizations - the paper explicitly asks whether these curves are rational when $\mathcal{P}$ is birational and answers negatively in general - and no criterion determines for which birational parametrizations they are rational. The problem asks for the effective completion of this resultant-based theory: explicit membership algorithms with correctness proofs; a procedure extracting individual base-point multiplicities from resultant data; and, for birational parametrizations, a rationality criterion for the auxiliary curves expressed in terms of the parametrization.

Background

A rational parametrization of an algebraic surface is a rational map $\mathcal{P}=(p_1:p_2:p_3:p_4)$ from the projective plane $\mathbb{P}^2$ to $\mathbb{P}^3$, given by four homogeneous polynomials of the same degree with no common factor, whose image is a dense subset of the surface $\mathscr{S}$. The points of $\mathbb{P}^2$ at which all four polynomials vanish simultaneously are the base points of the parametrization; for surface parametrizations the base locus $B(\mathcal{P})$ is finite (possibly empty). Base points control the quantitative geometry of a parametrization through the degree formula

$\deg(\mathcal{P})^2 = \operatorname{mult}(B(\mathcal{P})) + \deg(\mathscr{S})\,\deg_{\mathrm{Map}}(\mathcal{P}),$

where $\deg(\mathcal{P})$ is the common degree of the $p_i$, $\deg(\mathscr{S})$ is the degree of the image surface, $\deg_{\mathrm{Map}}(\mathcal{P})$ is the degree of the rational map (the cardinality of a generic fiber, computable by the algorithms of Pérez-Díaz and Sendra), and $\operatorname{mult}(B(\mathcal{P}))$ is the total multiplicity of the base locus. The standard local measure is the Hilbert-Samuel multiplicity: for a base point $A$, $e(I_A,R_A)$ is the multiplicity of the local ring of $\mathbb{P}^2$ at $A$ modulo the ideal $I_A$ generated by the $p_i$ (equivalently, the leading coefficient of its Hilbert-Samuel polynomial), and $\operatorname{mult}(B(\mathcal{P}))=\sum_{A\in B(\mathcal{P})} e(I_A,R_A)$. Each individual $e(I_A,R_A)$ can in principle be computed by local standard-basis (tangent cone) methods in a computer algebra system, but this is expensive, which motivated the search for global elimination formulas for the total.

Cox, Pérez-Díaz and Sendra (2020, published 2022) proved such a formula. Fix an algebraically closed field $\mathbb{K}$ of characteristic zero. After a projective change $L=(L_1:L_2:L_3:L_4)$ of the target coordinates of $\mathbb{P}^3$, form the two polynomials

$K_1^L = x_4 L_1(\mathcal{P}) - x_1 L_4(\mathcal{P}), \qquad K_2^L = x_4 L_3(\mathcal{P}) - x_3 L_4(\mathcal{P})$

in the parameter variables $t_1,t_2,t_3$ with coefficients in $\mathbb{K}[x_1,x_3,x_4]$, where the $x_i$ are new indeterminates. Then

$\operatorname{mult}(B(\mathcal{P})) = \deg_t\big(\operatorname{Content}_x(\operatorname{Res}_{t_3}(K_1^L,K_2^L))\big)$

for every $L$ in a non-empty Zariski open subset $\Omega$ of the projective transformation group $G(\mathbb{P}^3)$, where $\operatorname{Res}_{t_3}$ is the resultant eliminating $t_3$ — the classical determinant construction applied to the two polynomials viewed in the variable $t_3$, which vanishes exactly when they share a common $t_3$-root — and $\operatorname{Content}_x$ extracts the greatest common divisor of the coefficients in the $x$-variables (the remaining factor is the primitive part). The degree of the primitive part similarly recovers $\deg(\mathscr{S})\,\deg_{\mathrm{Map}}(\mathcal{P})$, and the argument yields an elementary proof of the degree formula, first proved in full generality by Cox (2001) using Segre classes from intersection theory (a device measuring the contribution of degeneracy loci to intersection numbers). Related earlier work connects base point multiplicities to implicitization — recovering the defining equation of a surface from its parametrization — via syzygies (polynomial relations among the components of a parametrization) and the moving surfaces they define (Busé, Cox and D'Andrea, 2003).

The proof, however, is generic in three ways that leave the method incomplete. First, the admissible transformations are described only existentially: the identities are proved on non-empty Zariski open subsets $\Omega$ and $\Omega'$ of $G(\mathbb{P}^3)$ (Lemma 1 and Lemma 2 of the paper) and, for the analogous formulas for dominant rational self-maps $\mathcal{S}$ of $\mathbb{P}^2$ (self-maps with dense image), on subsets $\Omega_{\mathcal{S}}$, $\Omega'_{\mathcal{S}}$, $\Omega''_{\mathcal{S}}$ of $G(\mathbb{P}^2)$ (Theorem 6, Theorem 7, Lemma 4); here a Zariski open subset means the complement of the common zero locus of finitely many polynomial conditions, so a non-empty one contains 'almost all' transformations, and although the proofs construct these sets through explicit non-vanishing conditions on the coefficients of $L$, the paper gives no algorithm that decides, for a concrete $L$, whether the identities apply. Second, the resultant formula computes only the sum of the Hilbert-Samuel multiplicities, not the individual values: the paper's Example 1 computes $\operatorname{mult}(B(\mathcal{P}))=4$ "without knowing anything about the number of base points or their individual multiplicities". Third, the elimination runs through auxiliary curves $\mathscr{C}(K_i^L)\subset\mathbb{P}^3$ defined over the transcendental field $\mathbb{F}=\overline{\mathbb{K}(x)}$ (an algebraic closure of the rational function field $\mathbb{K}(x)$): for birational self-maps $\mathcal{S}$ of $\mathbb{P}^2$ (rational self-maps admitting a rational inverse) the analogous curves $\mathscr{C}(J_i^L)$ are rational (Proposition 3 of the paper), but for surface parametrizations the paper exhibits a birational $\mathcal{P}$ (with $\deg_{\mathrm{Map}}=1$, $\operatorname{mult}(B(\mathcal{P}))=3$, $\deg(\mathscr{S})=6$) for which $\mathscr{C}(K_1^L)$ and $\mathscr{C}(K_2^L)$ have genus 1 for every $L$ in a non-empty open set, and no characterization of when rationality holds is given.

Problem Statement

Complete the effective theory of the resultant-based base point formula of Cox, Pérez-Díaz and Sendra for projective rational surface parametrizations $\mathcal{P}:\mathbb{P}^2\dashrightarrow\mathbb{P}^3$ over an algebraically closed field $\mathbb{K}$ of characteristic zero and, where the source paper develops the parallel formulas, for dominant rational self-maps $\mathcal{S}$ of $\mathbb{P}^2$, with no birationality or structural hypothesis beyond a finite base locus in deliverables 1 and 2; deliverable 3 concerns birational parametrizations, following the source's question. Three linked deliverables share one objective - turning the generic existence statements of the source paper into a deterministic, certified account of what the resultant data of a parametrization reveals about its base points.

1. Effective genericity. Give an explicit algorithmic characterization of admissible transformations for the formulas of the source paper - for the surface case the sets $\Omega$ (Lemma 1/Theorem 2) and $\Omega'$ (Lemma 2) of admissible $L\in G(\mathbb{P}^3)$, and for dominant rational self-maps $\mathcal{S}$ of $\mathbb{P}^2$ the sets $\Omega_{\mathcal{S}}$ (Theorem 6), $\Omega'_{\mathcal{S}}$ (Theorem 7) and $\Omega''_{\mathcal{S}}$ (Lemma 4). Since the source paper builds these sets from explicit non-vanishing, gcd and irreducibility conditions that involve choices (for instance $\Omega_1=\{L: B_1(z^L)B_2(z^L)\neq 0\}$ in equation (2.8), where $B_1,B_2$ are any non-zero coefficients of the auxiliary polynomials $G_1,G_2$), the characterization may be given for conditions extracted explicitly from the source proofs, or for any explicitly characterized Zariski open subset of the transformation group - defined by finitely many decidable polynomial conditions on $L$ - on which the corresponding resultant-content identity of the source paper holds; it need not reproduce the exact sets of the source paper. On input a parametrization $\mathcal{P}$ (and the map $\mathcal{S}$ in the self-map case) and a concrete projective transformation $L$ over a computable field of characteristic zero, membership must be decided in finitely many arithmetic steps. The procedure must be proved correct: every accepted $L$ satisfies the corresponding resultant-content identity of the source paper, and every rejected $L$ comes with a certificate that a specific one of the stated defining conditions fails for that $L$; the accepted set must contain a non-empty Zariski open subset on which the identity holds.

2. Individual base-point multiplicities. Give a general elimination-theoretic procedure that extracts each individual local multiplicity $e(I_A,R_A)$, $A\in B(\mathcal{P})$, from resultant, content and primitive-part data of explicitly constructed resultants - specifying which finitely many admissible transformations $L$ are used - together with a correctness proof.

3. Geometry of the auxiliary curves. For birational parametrizations $\mathcal{P}$ - the case in which the source paper asks whether the auxiliary curves $\mathscr{C}(K_i^L)\subset\mathbb{P}^3(\mathbb{F})$, $\mathbb{F}=\overline{\mathbb{K}(x)}$, "are also rational" and answers negatively in general via a genus-1 example - characterize, in terms of the parametrization $\mathcal{P}$, exactly when these curves are rational (genus 0) for generic $L$.

References

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