Background and relation to earlier work

The plane problem originates in Nagata’s work on Hilbert’s fourteenth problem. He conjectured a lower bound on the degree of a plane curve in terms of its multiplicities at very general points, and proved the strict inequality when the number of points is a square at least \(16\) (Nagata 1959). Multipoint Seshadri constants express the corresponding positivity question on an arbitrary polarized variety. The qualitative Nagata–Biran–Szemberg formulation asks for maximality at every sufficiently large point count. Roé and Ross state this formulation in arbitrary dimension and distinguish the ordinary curve-defined constant from its higher-cycle analogues (Roé and Ross 2009, Conjecture 1.3 and Remark 1.4). We use the ordinary constant throughout, even when \(\dim X>2\).

Biran’s packing stability theorem supplies an important symplectic precedent: sufficiently many equal balls can fill a closed symplectic four-manifold with rational symplectic class with arbitrarily small volume deficit (Biran 1999); see also (Biran 2001, Theorem 6.3). The algebraic question fixes the complex variety and asks for a nef class on its point blow-ups, so symplectic packing stability is not by itself the required algebraic statement. In arbitrary dimension, Küchle obtained lower bounds for multipoint Seshadri constants that are asymptotically optimal as the number of points tends to infinity (Küchle 1996, Theorem 1.1). Biran’s product inequality, extended to higher-cycle constants by Roé–Ross, relates point counts on a variety to point counts on projective space (Roé and Ross 2009, Theorem 1.1 and Remark 1.2). These results illuminate the volume scale in (2); an asymptotically optimal estimate alone does not imply eventual equality.

The companion paper (OpenAI 2026) establishes the qualitative surface statement by a nodal exponent bound, diagonal slice compression, and a primitive lattice direction tuned to the number of points. Its final tests are injective jet evaluations on finite torus subgroups. We adapt the planar geometric mechanism to polynomial spaces in two variables and use surjective full-dimensional jet evaluations instead. All arguments required for this adaptation are proved here; the surface maximality theorem is not an input.

The interpretation of positivity through asymptotic jet generation is classical; see Demailly (Demailly 1992, sec. 6). Our use of initial monomials belongs to the valuation approach to linear series initiated by Okounkov (Okounkov 1996) and developed by Lazarsfeld–Mustaţă (Lazarsfeld and Mustaţă 2009) and Kaveh–Khovanskii (Kaveh and Khovanskii 2012). More directly, Ito compares multipoint jet separation with initial monomial spaces by separate finite-degree degenerations, without a finite-generation hypothesis (Ito 2013, Proposition 5.7 and Theorem 5.8). Monomial diagrams and interpolation minors also underlie Dumnicki’s diagram-cutting method (Dumnicki 2007, Proposition 13 and Theorem 14). We retain elementary proofs of the specific transfer and counting statements used below.

Proof strategy and the two-coordinate construction

Separating jets through degree \(m\) at a tuple means prescribing, independently at each point, all Taylor coefficients of total degree at most \(m\), in local coordinates and line-bundle frames. If \(H^0(X,kL)\) separates these jets, then \[\varepsilon(X,L;\mathbf p)\ge \frac{m}{k};\] the curve-intersection argument is given in Lemma 14. Thus, for \(w=(V/r)^{1/n}\), our interpolation target is jet separation through degree \(\lfloor k\theta\rfloor\) for every fixed \(0<\theta<w\) and all sufficiently large \(k\). We first construct a tuple for each such test and later obtain simultaneous tests at very general tuples.

The argument works with finite exponent sets \(S_k\subset\mathbb Z^n\), one for each positive integer degree \(k\), satisfying \[S_k+S_l\subset S_{k+l}.\] For a set \(S\), its monomial space is the span of the Laurent monomials \(z^\alpha\) with \(\alpha\in S\) on \((\mathbb C^*)^n\). The multiplication condition will be as important as the geometry of the individual sets.

A complete-intersection flag on \(X\) first supplies sets \(S_k\) whose cardinalities are \(h^0(X,kL)\). They lie in the dilates of a simplex \[\Delta(a_1,\ldots,a_n) =\left\{x\in\mathbb R_{\ge0}^n:\sum_{i=1}^n\frac{x_i}{a_i}\le1\right\}\] of volume \(V/n!\). Jet surjectivity for the corresponding monomials can be transferred back to sections on \(X\).

The main construction replaces any two intercepts \(A,B\) of the bounding simplex by \(w,AB/w\), for any sufficiently small positive real \(w\). It preserves the number of exponents, the multiplication condition, and the implication from jet surjectivity after the replacement to jet surjectivity before it. The other exponent coordinates are carried along unchanged. The planar node bound and primitive-direction argument come from the surface method; the real-intercept replacement and its compatibility with these remaining coordinates allow the process to be iterated in higher dimensions.

For a fixed sufficiently large integer \(r\), choose \(w=(V/r)^{1/n}\). Repeated replacements produce sets \(F_k\) inside dilates of \[P_* = \Delta(w,\ldots,w,rw), \qquad \#F_k=h^0(X,kL), \qquad F_k+F_l\subset F_{k+l}.\] Because this simplex still has volume \(V/n!\), only \(o(k^n)\) of its degree-\(k\) lattice points are missing. The multiplication condition upgrades this density statement: for each fixed compact set \(K\subset\operatorname{int}P_*\), every lattice point of \(kK\) belongs to \(F_k\) for all sufficiently large \(k\). Indeed, such a point has on the order of \(k^n\) decompositions into two nearly equal degrees, whereas the holes can exclude only \(o(k^n)\) of them. This short argument is a full-density version of the interior approximation results for semigroups (Kaveh and Khovanskii 2012, Theorem 1.6).

The long last intercept of \(P_*\) permits explicit interpolation at \(r\) points that differ only in their last coordinate. One-variable Hermite interpolation, applied to the coefficients of monomials in the other coordinates, gives arbitrary jets through degree \(\lfloor k\theta\rfloor\) for every fixed \(0<\theta<w\) and all sufficiently large \(k\). Transferring these tests to \(X\), taking a countable intersection of nonempty Zariski-open loci, and then letting \(\theta\) increase to \(w\) proves Theorem 1.

Section 2 establishes the monomial operations and their jet transfers, and Section 3 constructs the initial exponent sets on \(X\). Section 4 proves the two-coordinate replacement. Section 5 establishes interior filling and interpolation, and Section 6 assembles the very-general conclusion.

Jets and graded exponent sets

Write \(\mathbb T^n=(\mathbb C^*)^n\). For a finite set \(S\subset\mathbb Z^n\), put \[\mathcal M(S)=\mathop{\mathrm{span}}_{\mathbb C}\{x^\alpha:\alpha\in S\}, \qquad x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}.\] These Laurent polynomials are regular on \(\mathbb T^n\). If \(A\) is a line bundle on a smooth complex variety \(Y\), its jets through an integer degree \(m\geq0\) at \(p\in Y\) form the vector space \[\mathrm J_p^m(A) =A_p/\mathfrak m_p^{m+1}A_p.\] Here \(A_p\) denotes the stalk of the sheaf of sections. A space of sections \(W\) separates jets through degree \(m\) at distinct points \(p_1,\ldots,p_r\) if the evaluation map \[W\longrightarrow\bigoplus_{i=1}^r\mathrm J_{p_i}^m(A)\] is surjective. For functions we use the trivial line bundle. Local frames and local coordinates change the matrices of these maps but not their ranks. Analytic and algebraic finite jets agree at a smooth complex point, since the corresponding local rings have the same completion. Our aim is to obtain such surjections for \(W=H^0(X,kL)\) with \(m/k\) approaching \((V/r)^{1/n}\).

We will say that an operation on monomial spaces transfers jet separation backward if, for every fixed degree, number of points, and jet order, separation by the output space at some distinct torus tuple implies separation by the input space at some distinct torus tuple. The tuples need not be the same. A finite sequence of such operations has the same property.

Least terms and analytic limits

A monomial order on \(\mathbb Z_{\geq0}^q\) will mean a total well-order preserved by addition. For a nonzero convergent power series \(f\), let \(\nu(f)\) be the least exponent in its support for this order.

The finite-degree passage to initial monomials is a standard jet-degeneration principle; see Ito (2013, Proposition 5.7). The next two lemmas give the direct analytic form used here, including unchanged coordinate factors. The first is analytic; the second is finite-dimensional linear algebra.

In particular, the unchanged variables in Lemma 5 cause no restriction on the jets being separated: all derivatives, including derivatives in those variables, enter the same convergent jet matrix. We shall apply the lemmas to finitely many spaces distinguished by a common exponent in the unchanged variables.

Lattice changes and coordinate compression

We next record two operations directly on Laurent monomials. They will let us change the shape of an exponent set while retaining both properties above.

For compression in the first coordinate, write an exponent as \((a,\beta)\) with \(a\in\mathbb Z\) and \(\beta\in\mathbb Z^{n-1}\). For a finite set \(S\), define its row space at \(\beta\) by \[W_\beta(S)=\mathop{\mathrm{span}}_\mathbb C\{z^a:(a,\beta)\in S\},\] and define the compressed exponent set by \[ \mathcal C_1(S)= \{(\mathop{\mathrm{ord}}_{z=1}f,\beta):0\ne f\in W_\beta(S)\}. \tag{5}\] All functions in a row are holomorphic near \(z=1\), even when their exponents are negative. Compression in another coordinate is defined by permuting coordinates.

When these operations act on a graded family, the support sets are defined by fixed exact orders and fixed lattice maps. The weights and small parameters used to transfer jets may nevertheless depend on the degree: they only establish finite-rank implications. The multiplication property in Lemma 4 holds independently of those analytic choices.

The exponent sets for every sufficiently large point count

Fix the polarized variety \((X,L)\) of Theorem 1, and write \(V=L^n\). Choose an integer \(D>0\) for which \(DL\) is very ample. Proposition 9 supplies additive support sets \(S_k\) with \[S_k\subset kP_0\cap\mathbb Z^n, \qquad \#S_k=h^0(X,kL).\] Here \(P_0=\Delta(1/D,\ldots,1/D,D^{n-1}V)\). Let \(\ell>0\) be the smallest intercept of \(P_0\). Choose an integer \(r_0\) such that \[ (V/r)^{1/n}<\ell/100\qquad(r\ge r_0). \tag{29}\] Such a choice exists because \(X\) and \(L\) are fixed.

Fix any integer \(r\ge r_0\), and put \(w=(V/r)^{1/n}\). Apply Proposition 10 to the first two intercepts, replacing them by \(w\) and their product divided by \(w\). Keep the intercept \(w\) fixed, and pair the other new intercept with the next untouched one. Continue until \(n-1\) intercepts equal \(w\).

Every pair of input intercepts \(A,B\) in this procedure satisfies \(A,B\ge\ell\). Indeed, an untouched intercept has this property, and the new carried intercept \(AB/w\) is greater than \(A\), since \(B\ge\ell>w\). Thus the hypothesis of Proposition 10 holds at every step: \[w\max(3/A,2/B)\le\frac{3w}{\ell}<\frac1{16}.\] The product of all intercepts remains \(V\). We obtain finite sets \(F_k\) in \[ kP_*,\qquad P_* =\Delta(w,\ldots,w,V/w^{n-1}) =\Delta(w,\ldots,w,rw), \tag{30}\] with \[F_k+F_l\subset F_{k+l}, \qquad \#F_k=h^0(X,kL)=\frac{V}{n!}k^n+o(k^n).\] Since \(\mathop{\mathrm{vol}}(P_*)=V/n!\), these sets satisfy the hypotheses of Proposition 13.

For each \(0<\theta<w\) and all sufficiently large \(k\), that proposition gives simultaneous jets through degree \(\lfloor k\theta\rfloor\) for \(\mathcal M(F_k)\) at an \(r\)-tuple of distinct torus points. Each application of Proposition 10 transfers this surjectivity to the preceding support set. Proposition 9 finally transfers it to \(H^0(X,kL)\) at some tuple of distinct points of \(X\). In particular, for every such \(\theta\) and \(k\), the set \[ \left\{\mathbf p\in U_r(X): H^0(X,kL)\longrightarrow \bigoplus_{i=1}^r\mathrm J_{p_i}^{\lfloor k\theta\rfloor}(kL) \text{ is surjective}\right\} \tag{31}\] is nonempty. The tuple witnessing this assertion may depend on both \(\theta\) and \(k\).

A common very-general locus

For fixed integers \(k\ge1\) and \(m\ge0\), the locus of tuples at which \(H^0(X,kL)\) surjects onto jets through degree \(m\) is Zariski open in \(U_r(X)\). Indeed, the bundle of principal parts of order \(m\) of \(kL\) is locally free because \(X\) is smooth. Pulling it back along each coordinate projection of \(U_r(X)\) gives an algebraic vector-bundle map \[H^0(X,kL)\otimes\mathcal O_{U_r(X)} \longrightarrow \bigoplus_{i=1}^r\operatorname{pr}_i^*\mathcal P^m(kL).\] Its surjectivity locus is defined by the nonvanishing of maximal minors.

Choose \(0<\theta_j<w\) with \(\theta_j\uparrow w\). For each \(j\), choose \(k_j\) so that the locus in (31) is nonempty and \(\lfloor k\theta_j\rfloor\ge1\) for every \(k\ge k_j\). Denote this open locus by \(O_{j,k}\) and set \[ Z_r=\bigcup_{j\ge1}\ \bigcup_{k\ge k_j} \bigl(U_r(X)\setminus O_{j,k}\bigr). \tag{32}\] Every set in this countable union is a proper Zariski-closed subset of \(U_r(X)\).

The complement is nonempty. The complex points of \(U_r(X)\) form a nonempty locally compact Hausdorff space, and hence a Baire space. Since \(U_r(X)\) is smooth and irreducible, every proper algebraic closed subset has empty interior in its complex topology. A countable union of these closed subsets cannot cover \(U_r(X)\). Thus every tuple outside \(Z_r\) simultaneously has all the jet-surjectivity properties indexed by \(j\) and \(k\ge k_j\).

From jets to ordinary curve multiplicities

The relation between Seshadri constants and jet separation appears in Demailly’s one-point formulation (Demailly 1992, sec. 6) and in the weighted multipoint framework of Ito (2013, sec. 5). The following elementary implication is the one needed here. We give its local proof, including at singular points of the curves.

Fix a tuple \(\mathbf p\in U_r(X)\setminus Z_r\). Applying Lemma 14 with \(A=kL\) and \(m=\lfloor k\theta_j\rfloor\) gives \[\varepsilon(X,L;\mathbf p)\ge\frac{\lfloor k\theta_j\rfloor}{k} \qquad(k\ge k_j).\] First let \(k\to\infty\) for each fixed \(j\), and then let \(j\to\infty\). We obtain \[ \varepsilon(X,L;\mathbf p)\ge w=(V/r)^{1/n}. \tag{34}\]

The volume bound

For completeness, we derive the reverse inequality for every tuple of distinct points. Let \(\pi:Y\to X\) be the blow-up of such a tuple, with exceptional divisors \(E_1,\ldots,E_r\). For \(t\ge0\), put \[D_t=\pi^*L-t\sum_{i=1}^r E_i.\] If an integral curve \(\Gamma\) is contained in \(E_i\simeq\mathbb P^{n-1}\), then \(D_t|_{E_i}=t\mathcal O_{\mathbb P^{n-1}}(1)\) is nonnegative on \(\Gamma\). Every other integral curve is the strict transform of an integral curve \(C\subset X\), and \[ D_t\cdot\Gamma=L\cdot C-t\sum_{i=1}^r\mathop{\mathrm{mult}}_{p_i}C. \tag{35}\] Here \(E_i\cdot\Gamma=\mathop{\mathrm{mult}}_{p_i}C\) also when \(C\) is singular. If \(p_i\notin C\), both sides vanish. For \(p_i\in C\), identify the strict transform locally with the blow-up of \(\mathcal O_{C,p_i}\) at its maximal ideal. Its intersection with \(E_i\) is the zero-dimensional exceptional fiber \(\operatorname{Proj}(\operatorname{gr}_{\mathfrak m}\mathcal O_{C,p_i})\), whose length is the Hilbert–Samuel multiplicity. This is the degree of the effective Cartier divisor cut out by \(E_i\) on the strict transform.

It follows from (35) that \(D_t\) is nef whenever \(0\le t<\varepsilon(X,L;\mathbf p)\). A nef real divisor has nonnegative top self-intersection. The exceptional divisors are disjoint, \(\pi^*L|_{E_i}\) is trivial, and \(E_i^n=(-1)^{n-1}\). Therefore \[0\le D_t^n=V-rt^n.\] This inequality for all \(0\le t<\varepsilon(X,L;\mathbf p)\) proves the universal bound \[ \varepsilon(X,L;\mathbf p)\le(V/r)^{1/n}. \tag{36}\] Together, (34) and (36) give the equality asserted in Theorem 1 at every tuple outside \(Z_r\). Equation (35) also shows that \(D_w\) is nef there, and its top self-intersection is zero. The threshold (29) was chosen before fixing \(r\), so this proves the theorem for every integer \(r\ge r_0\).