Nagata’s conjecture for plane curves

1 Introduction

Prescribing multiple points of a plane curve is an interpolation problem: how small can its degree be when the positions and the required orders of vanishing are given? Nagata’s conjecture predicts the answer at the square-root scale for sufficiently many very general points. Nagata introduced the problem in his work on Hilbert’s fourteenth problem and proved the square cases r\ge16 [12, 13]. We prove the strict inequality for every r\ge10, allowing unequal multiplicities.

We first work over \C. Let

A nonzero effective plane curve is the zero divisor of a nonzero homogeneous form of degree d\ge1. Its multiplicity at a point is the order of that form in the regular local ring. Components may repeat. The phrase very general in the following theorem means the complement of one countable union of proper closed subsets, chosen simultaneously for all degrees and multiplicity vectors.

The scale can already be seen when all prescribed multiplicities equal m. A degree-d form has \binom{d+2}{2} coefficients, and order at least m at r points imposes at most r\binom{m+1}{2} linear conditions. Their leading terms balance at d=m\sqrt r. A count of conditions supplies curves asymptotically above this scale; proving their absence below it requires control of the dependencies among the conditions, for every m. The strict endpoint also matters: at nine points a nonzero cubic always exists, so the asserted strict inequality cannot start at r=9.

On the blowup of the plane at such a tuple, the theorem gives a real divisor class of self-intersection zero that intersects every curve positively. It also determines the multipoint Seshadri constant: the largest t for which the pullback of a line minus t times the sum of the exceptional curves has nonnegative intersection with every curve. Its value is 1/\sqrt r, which is irrational when r is nonsquare. Section 6 proves these consequences and extends the theorem to every uncountable algebraically closed field of characteristic zero.

1.1 Prior work and the interpolation approach

The geometry of plane linear systems connects this problem with the cones of effective and nef divisor classes on blowups. Ciliberto–Harbourne– Miranda–Roé [4] develop these connections and compare several forms and extensions of Nagata’s conjecture. For homogeneous multiplicities, Roé [15] obtained the uniform estimate d>m(\sqrt{r-1}-\pi/8) for r\ge10 by specializing to configurations with infinitely near points. Harbourne [11] sharpened rational lower bounds and proved further cases in restricted multiplicity ranges. For ten very general points, degeneration methods give d/m\ge117/37 for every nonempty homogeneous system: Eckl’s Seshadri estimate and Ciliberto–Dumitrescu–Miranda–Roé’s analysis of plane systems give this bound in their respective formulations [9, 3]. Such bounds explain the importance of controlling arbitrarily large multiplicities at each fixed nonsquare r.

Dumnicki–Harbourne–Küronya–Roé–Szemberg [8] study an analogue for very general monomial valuations, while Nivoche [14] relates the conjecture to transcendental interpolation in \C^n. These perspectives emphasize that the obstruction is not merely the number of imposed conditions, but how vanishing conditions interact with the geometry of the ambient surface. More recently, Ciliberto–Miranda–Roé [6, Theorem 1] constructed irrational square-zero nef rays on the blowups at every r\ge10 very general points. Those constructions do not determine the equal-weight multipoint Seshadri constant. Nagata’s conjecture singles out the class with equal exceptional coefficients, whose nefness gives that constant’s exact value.

The proof below uses two established approaches to this interaction. Ciliberto–Miranda [5] develop degenerations of plane linear systems, and Evain [10] studies their infinitesimal limits when fat points, meaning points equipped with specified orders of vanishing, approach a divisor. Dumnicki [7] encodes interpolation by finite diagrams of monomial exponents and their evaluation matrices. In particular, the proof of his Proposition 13 converts a one-point derivative matrix into polynomial evaluation on the exponent diagram. Our final rank problem has the same evaluation interpretation, with a different set of allowed polynomial degrees supplied by a collinear collision.

The elliptic stage uses the classical multiplicative theta product and its residue-class Laurent bases; the conventions used here are recorded in Rosengren [16, arXiv version, Section 1.2 and Exercise 1.6.5]. We prove the needed product identities, basis formulas and convergence estimates in the normalization required by the argument.

1.2 Proof strategy

Closed incidence loci turn a failure at very general points into a system of curves existing at every configuration. A product over cyclic permutations makes the multiplicities equal; write d and m for the resulting degree and common multiplicity. Moving the points toward a smooth curve gives a nonzero polynomial on its normal line bundle, with order at least m at arbitrary normal displacements. For square r, its two highest coefficients already exclude d/m\le\sqrt r, including equality (Section 2).

For nonsquare r, we use a cubic and set nine normal displacements to zero. Dividing out the forced coefficient zeros and changing the fiber coordinate give a finite-dimensional space of polynomial sections whose first m+1 coefficient bundles have equal degree (Section 3). On each fixed elliptic curve, the remaining q=r-9 points collide along a horizontal coordinate line. After b<m fiber derivatives, a section must then have base order at least q(m-b); thus a particular finite jet map has a nonzero kernel. A separate degeneration gives explicit normalized theta bases whose expressions in logarithmic base and fiber coordinates (x,y) tend to e^{Kx+jy}. Here j is the fiber degree and K a Laurent exponent. Their mixed derivatives give the limiting matrix

with columns indexed by exponent pairs (j,K) (Section 4). Only these numerical matrices are compared across curves; the argument does not choose a simultaneous family of kernel vectors.

To exclude the kernel, a total count of exponent pairs is not enough. We group them into horizontal rows of fixed K, with j varying. Interpolation one row at a time gives full column rank if no row has more than m points and at most q(m-t+1) rows have at least t points. The exponent pairs lie in a polygon whose slice-width profile supplies exactly these bounds. Since d/m is rational and r is nonsquare, the strict gap d/m<\sqrt r permits a contraction of that polygon; taking a common power of the original curve absorbs the integer endpoint error in the row count. Full rank persists for small positive degeneration parameter, contradicting the kernel on each fixed curve (Section 5).

2 From plane curves to normal polynomials

We reduce a violation at very general points to an equal-multiplicity system existing at every configuration. Moving its points toward a smooth plane curve then gives a polynomial on the normal bundle. This polynomial will exclude the square cases and supply the input for the cubic construction in Section 3.

The order of a section is the order of its scalar expression in a local frame. A frame change multiplies this expression by a unit, preserving the order. In local coordinates, order at least m means that every Taylor coefficient of total degree less than m vanishes. Thus algebraic and holomorphic orders agree, and holomorphic coordinate changes preserve them.

2.1 A universal equal-multiplicity system

For positive integers d,m, write \mathsf U_r(d,m) for the property

Here and below the multiplicity of a form is the multiplicity of its zero divisor.

2.2 Specialization to a normal bundle

Let B\subseteq\Pp^2 be a smooth irreducible plane curve of degree k\ge1, and fix a nonzero homogeneous equation G_B for it. Put

The equation identifies the normal bundle of B in \Pp^2 with M, as in the formula for an effective Cartier divisor [17, Tag 0B3P]. In a local frame e for \mathcal O_{\Pp^2}(1), the local equation t=G_B/e^k identifies a normal vector with its derivative under t. For w\in M_a, the expression f_j(a)w^j is an element of L_a^d when f_j is a section of L^{d-kj}. Thus the polynomial in the next proposition is intrinsically a section of the pullback of L^d to \Tot(M).

2.3 The square case

We use two basic facts about smooth projective curves. A nonzero section of a line bundle has an effective zero divisor whose degree is the degree of the bundle. In particular, a degree-zero line bundle has a nonzero section only if it is trivial. Also, for a smooth complex projective curve of positive genus, \Pic^0(B) is a complex torus; its subgroup of elements of any fixed finite order is finite, so its torsion subgroup is countable [17, Tag 0C1Z].

Together with Lemma 2.1, the proposition proves Theorem 1.1 for square r, including the strict inequality and unequal multiplicities.

2.4 Nine zero displacements on a cubic

For the remaining argument we will apply Proposition 2.2 with k=3 and with zero normal displacements at nine distinct points of a smooth plane cubic. Write P=a_1+\cdots+a_9. Extend the coefficient list by f_j=0 for j>\lfloor d/3\rfloor. At each (a_i,0), expansion in the fiber variable shows that multiplicity at least m implies

Indeed, in a local base coordinate \zeta centered at a_i, the coefficient of \zeta^a w^j must vanish whenever a+j<m. Thus f_j has base order at least m-j at every point of P. The coefficient bundles in (2.3) all have degree 3(d-3m), independent of j. These conditions hold regardless of the additional points and normal displacements chosen in the proposition.

3 Polynomial sections on an elliptic curve

Nine zero normal displacements impose zeros on the coefficients of a normal polynomial. Dividing out these zeros puts its first m+1 coefficients in bundles of the same positive degree. We first make this change intrinsically on any smooth cubic. We then choose cubics and marked points for which theta functions give explicit bases of the resulting spaces, ready for the collision and limit in Section 4.

In this section and Section 4, H^0 denotes holomorphic sections. The algebraic sections supplied by Proposition 2.2 belong to these spaces, so excluding holomorphic solutions will also exclude the necessary normal polynomials.

3.1 Removing the nine prescribed coefficient zeros

For this construction, let d,m be any positive integers with d>3m. Let B be a smooth plane cubic, let L=\mathcal O_B(1) and M=L^3, and choose nine distinct points with divisor P=a_1+\cdots+a_9. Write \sigma_P for the canonical section of \mathcal O_B(P) with zero divisor P, and put

Thus \deg A=0 and \deg T=3(d-3m)>0. Multiplication by \sigma_P defines a bundle map A\to M, written

which is invertible over B\setminus P. Define coefficient spaces inside H^0(B,T\otimes A^{m-j}) by

Negative powers denote dual powers. The target in the second line is correct because

Multiplication by \sigma_P^{j-m} is injective, as can be checked on B\setminus P. Let \W be the space of polynomial sections on \Tot(A)

valued in the pullback of T\otimes A^m=L^d(-mP). It is a finite-dimensional space: its coefficient blocks are spaces of sections on the compact curve B, and distinct powers of the fiber coordinate make their sum direct.

The nine zero displacements are now built into \W. The remaining points impose ordinary multiplicities at scalar fiber coordinate 1, as follows.

We next realize these coefficient spaces on explicit elliptic curves. Their bases will retain the bundle multipliers, not merely their degrees; those multipliers determine the Laurent exponents in the limit.

3.2 Theta functions and plane cubics

For 0<\tau<1, let

This is a connected compact complex torus of dimension one. For n\in\Z and \gamma\in\C^*, define D(n,\gamma) as the holomorphic line bundle obtained from \C^*\times\C by the action generated by

In its covering-space frame, sections are holomorphic functions with

and ordinary multiplication gives the tensor identifications

Define

We write \Theta when \tau is fixed. On every compact annulus, the sums of the absolute deviations of the factors from 1 are bounded by geometric series. The products therefore converge locally uniformly on \C^*, and reindexing gives

Its zeros are precisely the points of \tau^{\Z}, all simple: exactly one factor vanishes at each, while the remaining product is nonzero. Thus \Theta(z/a) is a section of D(1,-a) with divisor the single point represented by a\in\C^*. Products of n such sections, with the parameters having product (-1)^n\gamma, show that D(n,\gamma) has degree n for n>0. Tensor products and duals give the formula for every integer n; in degree zero, use D(0,\gamma)=D(1,\gamma)\otimes D(1,1)^{-1}.

The multiplicative theta convention and residue-class bases are classical; see Rosengren [16, arXiv version, Section 1.2 and Exercise 1.6.5]. We include the proof to fix all multipliers and normalizations.

3.3 Choosing the cubic and marked points

We now fix a nonsquare integer r\geq10 and a hypothetical universal system \mathsf U_r(d,m) with d/m\leq\sqrt r. The rationality of d/m and Lemma 2.4 give 3<d/m<\sqrt r. Set

Put

Set x_0=1/2 and choose a rational number \Delta such that

The strict slope gap leaves an interval of such choices. Rationality of \Delta will keep the Laurent exponent intervals away from integral endpoints.

Choose nine distinct real numbers x_i with

For example, take x_i=x_0-i\Delta/45. The quantities \rho,\Delta,x_i,\eta can be kept unchanged under a common multiple of (d,m), whereas h,u_0 and the coefficient spaces depend on the actual pair. We keep this pair fixed while taking the collision on each curve and the subsequent limit in \tau.

For each 0<\tau<1, use Lemma 3.3 to identify X_\tau with a plane cubic B, with the holomorphic identification

Let a_i be the point represented by z_i and put P=a_1+\cdots+a_9. These points and the point represented by z_0 are distinct: their exponents lie in (0,1), and two lifts represent the same point only if their exponents differ by an integer.

In the covering-space frame, the section

has divisor P and gives an identification

under which the canonical section \sigma_P is represented by \Pi(z). Thus the bundles already used in the construction are

Since 3\eta=\sum_i x_i+\rho_*, we have

These identifications use the covering-space frames and their tensor products. We retain the displayed powers of \tau as written, without replacing the multiplier presentations by isomorphic ones.

A cubic equation identifies the normal bundle of B with M=L^3. Its differential at each point surjects onto the normal fiber, so every scalar specified in the chosen covering-space frame is an allowed normal displacement in Proposition 2.2. Rescaling the cubic equation does not restrict those choices. All constructions are made for each fixed \tau; their algebraic dependence on \tau is not required.

Denote the resulting polynomial space by \W_\tau. By Lemma 3.2, its first m+1 coefficient blocks have degree and dimension h. For j>m, the dimension is 3(d-3j) when d-3j>0, and is one when d-3j=0. In the last case the source is H^0(B,\mathcal O_B)=\C, since B is compact and connected.

Near the point represented by (z,v)=(z_0,1), use coordinates

Choose the x-disk small enough to map injectively to B and avoid P, and the y-disk small enough that e^y is injective. In the fixed covering-space frame of T\otimes A^m, the scalar expression is

Here g_j(z) denotes the holomorphic function in its covering-space frame; all summands take values in the same bundle.

For every fixed \tau\in(0,1) and every q distinct positions \xi_1,\ldots,\xi_q sufficiently near zero, apply Proposition 3.1 in this disk and the fixed frame of A. It gives a nonzero \widetilde F\in\W_\tau of multiplicity at least m at each (x,y)=(\xi_\alpha,0), since v=1 corresponds to y=0. The existence is pointwise in \tau and in the tuple of positions; the collision will use no simultaneous choice of sections.

4 Colliding points and degenerating sections

We turn the multiplicities at the remaining q points into derivative conditions at one point of \Tot(A). Universal existence will force these conditions to have a nonzero solution in \W_\tau. We then choose bases whose derivative matrices have an explicit limit as \tau\to0^+. Section 5 will prove full column rank when the common multiplicity is sufficiently large at a fixed ratio d/m.

All choices from Section 3, including d,m,q,h,u_0 and \rho,\rho_*,\Delta,x_i,x_0,\eta, remain fixed. The collision takes place with \tau fixed, before the matrix limit in \tau.

4.1 Collision on a fixed surface

The collision below belongs to the broader study of limits of fat-point conditions; compare Evain [10, Sections 2 and 6]. We prove the particular local constraint directly.

For each fixed \tau, use the coordinates and the frame in (3.7). If F\in\W_\tau has coefficients g_j, write its scalar expression in this chart as

Define the finite index set

and the linear map J_\tau:\W_\tau\longrightarrow\C^{\mathcal I} whose coordinates are

The value bundle is expressed in one fixed local frame when these derivatives are taken.

The rows in (4.1) describe a local ideal. If the distinct centers are (\xi_a,0) and Q(x)=\prod_{a=1}^q(x-\xi_a), write a local holomorphic function as \sum_{b\geq0}h_b(x)y^b. The multiplicity conditions require Q^{m-b} to divide h_b for b<m. When the centers collide, the corresponding limiting conditions are divisibility by x^{q(m-b)}, or membership in (x^q,y)^m. Their number is

Lemma 4.1 establishes these conditions for a nonzero limit. It remains to compute the matrix of J_\tau in bases suited to \tau\to0^+.

4.2 A common chart and normalized theta sections

Although the tori vary, one polydisk in (x,y) is a valid chart for all sufficiently small positive \tau. Fix R with 0<R<\pi/2, and let

For small enough \tau, require |\log\tau|>2R and \delta_P|\log\tau|>R. If z_0e^x and z_0e^{x'}, with |x|,|x'|<R, represent the same point of X_\tau, then

The bounds on the real and imaginary parts force a=b=0. Thus this disk maps injectively to X_\tau. If a point of the disk represented a point of P, its real coordinate would satisfy

whose absolute value is at least \delta_P|\log\tau|>R. Hence the disk avoids P. The covering-space frame trivializes A there, and v=e^y is injective for |y|<R. Thus (3.7) gives the common polydisk |x|,|y|<R around the marked point on \Tot(A). We continue to use the covering-space frames of Section 3.

4.3 Bases for the polynomial spaces

We now apply Lemma 4.2 to the actual coefficient spaces of \W_\tau. For 0\leq j\leq m, their bundle is

Since \eta-3x_0=(\rho_*-\Delta)/3, its interval parameter is

For m<j\leq u_0 with d-3j>0, the space in (3.1) is the image of the sections of

under multiplication by \Pi^{j-m}. The interval parameter for this source bundle is

These parameters are irrational. Indeed, after substituting \rho_*=3(\sqrt r-3), the coefficient of \sqrt r in either expression is the positive integer d-3j, and the remaining terms are rational because \Delta\in\mathbb Q. In particular,

The finite exponent set is

All indices j in this definition are integers. The last set is empty unless 3 divides d, and otherwise consists of the single pair (d/3,0). Because d>3m, that pair has j>m.

For a pair (j,K) in the first union, choose the normalized section from Lemma 4.2 in H^0(B,T\otimes A^{m-j}) as the coefficient of v^j. For a pair in the second union, choose the normalized section in H^0(B,L^{d-3j}), multiply it by \Pi^{j-m}, and use the result as the coefficient of v^j. For the last union, use \Pi^{j-m} as the coefficient of v^j: the source space is H^0(B,L^0)=\C, with basis the constant section 1. Denote the resulting fiber polynomials by E_{j,K,\tau}.

Each choice gives a basis of its coefficient block. For j>m, multiplication by \Pi^{j-m} is injective, since \Pi is nonzero on B\setminus P, and its image is the block by definition. The different powers v^j form a direct sum. Hence

is a basis of \W_\tau for every \tau\in(0,1). There are no duplicate pairs, and the index set is independent of \tau.

The block dimensions also give the useful identity

To see this, sum the dimensions 3(d-3j) of H^0(B,L^{d-3j}) for d-3j>0 and add one when 3 divides d. The result is \binom{d+2}{2}. For j<m, the coefficient condition in (2.3) reduces the block dimension by 9(m-j); their sum is 9\binom{m+1}{2}.

In the common local chart, the scalar expressions of this basis satisfy

locally uniformly, together with all fixed mixed derivatives. For the first union this is (4.2) times e^{jy}. For the second union, the additional factor \Pi(z_0e^x)^{j-m} tends to 1 with all derivatives by Lemma 4.2; the exponent j-m is fixed. The same product assertion proves the claim for the degree-zero case with K=0. As \mathcal S is finite, all of these convergences can be taken on one fixed smaller polydisk and for the same sufficiently small values of \tau.

4.4 The limiting jet matrix

Evaluation of monomials in exponent pairs is a useful way to express interpolation rank. Dumnicki [7, proof of Proposition 13] uses this principle to relate a one-point multiplicity matrix to polynomial evaluation on a diagram. Here the normalized theta sections produce the exponent pairs directly, with the derivative indices prescribed by the preceding collision.

The factors \tau^{-x_0K} may tend to zero or infinity, but for each \tau>0 they give invertible basis changes and preserve rank. Lemma 4.1 supplies a nonzero kernel separately for each fixed \tau; Proposition 4.3 compares numerical matrices in the explicit bases. Thus no continuous choice of kernel vectors, or algebraic family of plane embeddings across \tau=0, is required. To contradict Lemma 4.1, it remains to prove full column rank of the corresponding limiting matrix for a sufficiently large common multiple of (d,m).

5 Interpolation on the exponent set

We prove that the limiting matrix (4.5) has full column rank when the common multiplicity is sufficiently large at a fixed ratio d/m. The final assembly will achieve that size by taking a common power. The matrix columns are indexed by the finite exponent set \mathcal S of (4.4), and its rows are the restrictions to \mathcal S of the monomials

where evaluation at (j,K) substitutes J=j. Thus full column rank means that their span interpolates every function on \mathcal S. We first give a sufficient condition in terms of the distribution of horizontal row sizes. We then prove that condition by enclosing \mathcal S in a polygon and counting its horizontal slices. All parameters remain fixed as in Section 3; in particular, q=r-9 and h=m\rho>0.

5.1 Interpolation by horizontal rows

The interpolation procedure treats larger rows first and multiplies each correction by factors vanishing on all previously treated rows. The number of earlier rows controls the degree in K, while the current row size controls the degree in J. Their joint constraint explains why a bound on the largest row, or on the total number of points, would not suffice.

For the exponent set \mathcal S, it remains to prove the two row bounds in this lemma. We will obtain them by enclosing \mathcal S in a polygon and using its horizontal slice lengths to count the rows.

5.2 An enclosing polygon

Write a=\rho_*=3(\sqrt r-3)>0, and set

Thus U_0 is the lower endpoint of the exponent interval for j=0. For 0<\lambda\le1, and with t_+=\max\{t,0\}, define

By (3.5), we have \rho+\Delta\rho/9<a. Choose 0<\lambda<1 sufficiently close to 1 that

This choice depends only on r, the ratio d/m, and \Delta. We will show that this inequality places \mathcal S inside m\mathcal D_\lambda+(0,U_0). The horizontal widths of this scaled, translated polygon will be at most m\lambda<m, and the heights supporting a slice of length at least s will form an interval of length q(m\lambda-s) for 0\le s\le m\lambda.

5.3 Horizontal slices and integer rows

The length of a bounded interval is the difference of its endpoints; a singleton has length zero. When a required length is zero, we still require the slice to be nonempty.

For each integer K, let

Only finitely many of these row sizes are nonzero. The slice profile in Figure 1 bounds both each row size and the number of rows of any given minimum size. The passage from lengths to integer counts adds at most one endpoint. We will absorb it with the size condition

5.4 Full rank and the strict inequality

6 Fields and multipoint positivity

The analytic proof was carried out over \C. Its conclusion extends by algebraic incidence conditions, without embedding another field into \C. The implication from complex Nagata to the corresponding statement over uncountable algebraically closed characteristic-zero fields is also discussed by Bansal–Majumder [1]. We give the incidence argument with the simultaneous quantifiers needed here.

Let X be the blowup of the plane at a tuple covered by Theorem 1.1 or Corollary 6.1. Write H for the pullback of a line and E_1,\ldots,E_r for the exceptional curves. A real divisor class is nef if its intersection with every irreducible curve is nonnegative, and strictly nef if every such intersection is positive. The multipoint Seshadri constant of the line bundle \mathcal O_{\Pp^2}(1) at the tuple is