On OpenAI's proof of the Nagata conjecture

Introduction

Nagata’s conjecture grows out of the elementary Lagrange interpolation problem. We work over $\mathbb{C}$. Let $x_1,\ldots,x_r$ be distinct numbers. We are looking for a polynomial $f(x)$ with $f(x_i)=c_i$ for every $i$, where $c_1,\ldots,c_r$ are prescribed values. There is a unique solution of degree at most $r-1$:

The following lemma generalizes Lagrange interpolation. It will be used in subsequent sections.

Proof. Put $Q_i(y)=\prod_{0\leq t<i}(y-\beta_t)$ for $0\leq i\leq s$, so that $Q_0=1$. We construct an interpolating polynomial of the form $F=\sum_{i=0}^{s}Q_i(y)h_i(x)$ by choosing $h_0,\ldots,h_s$ successively with $\deg h_i<N_i$. Adding $Q_i(y)h_i(x)$ preserves the values on rows labeled $0,\ldots,i-1$, since $Q_i$ vanishes there. On row $i$, the nonzero factor $Q_i(\beta_i)$ reduces the remaining conditions to Lagrange interpolation in $x$ at $N_i$ distinct first coordinates, determining $h_i$ uniquely. The polynomials $x^aQ_i(y)$, $0\leq a<N_i$, form a basis of $V_I$: their leading terms in $y$ are $x^ay^i$, and the remaining monomials are indexed by $I$ because $N_t\geq N_i$ for $0\leq t<i$. This proves existence and uniqueness. $\square$

Given points $P_1,\ldots,P_r\in\mathbb{C}^2$, prescribing values $F(P_i)=c_i$ for a polynomial $F$ of degree at most $d$ gives a system of linear equations in its coefficients, generally inhomogeneous. We now ask whether the associated homogeneous system $F(P_i)=0$ has a nonzero solution. Equivalently, we seek a plane algebraic curve $F(x,y)=0$ passing through these points.

The polynomials of total degree at most $d$ form a vector space with basis $x^ay^b$, where $a,b\geq0$ and $a+b\leq d$. Its dimension is

Each condition $F(P_i)=0$ is one homogeneous linear equation in the coefficients. The corresponding $r\times D$ matrix is called the evaluation matrix; its entry in row $i$ and the column indexed by $x^ay^b$ is the value of the monomial $x^ay^b$ at $P_i$. If $r<D$, there are fewer equations than unknowns, so a nonzero solution exists for every configuration of points. The converse requires a condition on their position: arbitrarily many points can lie on a curve of fixed positive degree. To prove that this system has the largest possible rank for general points, apply Lemma 1.1 to the triangular set

Its row lengths are $d+1,d,\ldots,1$, so $I=T_d$. Thus evaluation at these $D=\binom{d+2}{2}$ points is an isomorphism on polynomials of degree at most $d$. The diagrams below show $d=3$ and $D=10$.

The entries of the evaluation matrix are polynomials in the point coordinates. For $r \le D$, choose $r$ points of $T_d$; for $r \ge D$, add $r-D$ distinct points to $T_d$. In either case, the evaluation matrix has rank $\min(r,D)$, so one of its minors of this size is nonzero. This minor remains nonzero outside the zero set of a nonzero polynomial in the point coordinates. A property that holds outside such a set is said to hold for general points. The kernel therefore has dimension $\max(D-r,0)$, proving

there is a nonzero polynomial of degree at most $d$ vanishing at $r$ general points $\Longleftrightarrow r < \binom{d+2}{2}$.

We can also prescribe values of the partial derivatives. Prescribing a function’s values and first derivatives, for example, determines the tangent planes to its graph. These are again linear conditions on the polynomial’s coefficients. As before, we study the homogeneous system of linear equations, in which the prescribed values and derivatives are zero.

Fat points. A polynomial $F$ vanishes to order at least $m$ at $P=(x_0,y_0)$ if

The ideal $(x-x_0,y-y_0)^m \subset\mathbb{C}[x,y]$ consists of the polynomials satisfying these conditions and defines the fat point of multiplicity $m$ at $P$. Evaluation at this fat point records the value of the polynomial along with all of its partial derivatives of order less than $m$. Equivalently, it records the class of $F$ in the quotient ring $\mathbb{C}[x,y]/(x-x_0,y-y_0)^m$, which is the coordinate algebra of the fat point. This ring has dimension $\binom{m+1}{2}$ over $\mathbb{C}$, which is called the length of the fat point. The length is equal to the number of rows a fat point contributes to the evaluation matrix.

For a second proof of (2), take the fat point of multiplicity $d+1$ at the origin. Its length is $D=\binom{d+2}{2}$, and evaluation on polynomials of degree at most $d$ is an isomorphism: it simply records all coefficients of these polynomials. By Lemma 2.1, this fat point admits a flat deformation into $D$ distinct points. The entries of the evaluation matrix are regular in the deformation parameter. Its determinant is nonzero at the fat point and hence remains nonzero for nearby parameters. Evaluation at the resulting $D$ distinct points is therefore an isomorphism. For $r<D$, take a subset of these points; for $r>D$, add distinct points. The evaluation matrix has rank $\min(r,D)$ in either case, and the same is true for general points by the argument with minors above.

Imposing multiplicity $m$ at each of $r$ distinct points gives $r\binom{m+1}{2}$ homogeneous linear conditions on the coefficients. Thus, for every configuration of points, a nonzero polynomial of degree at most $d$ vanishing to the prescribed orders exists whenever

If this inequality fails, a nonzero solution may still exist because the equations may be linearly dependent. For fixed $r$, as $d$ and $m$ grow proportionally, the leading terms are $d^2/2$ and $rm^2/2$. This suggests $d/m=\sqrt{r}$ as the boundary for the existence of such curves.

Here very general means outside a countable union of proper algebraic subsets of the configuration space. This allows the inequality to hold for every pair of integers $d$, $m$ at the same configuration of points. The restriction $r \ge10$ is necessary: nine points always lie on a nonzero cubic. Nagata proved the conjecture when $r$ is a square [15, 16]; see [3 §2] for its history and related questions.

Scope of these notes. I began writing these notes while preparing my talk at UMass Amherst. They are not intended as a definitive account, and I claim no originality. I have reorganized and simplified the argument by working directly on the normal ruled surface of a smooth plane cubic, without performing elementary transformations of ruled surfaces. § 3 explains interpolation duality and proves the interpolation statement on a triangle of lattice points. § 4 realizes this interpolation matrix as a limit of matrices obtained by evaluating theta functions and their derivatives. § 5 transfers the problem to the plane and proves Nagata’s conjecture. § 6 applies the same construction to monomial valuations and determines the nef and pseudoeffective cones of weighted blow-ups of $\mathbb{P}^2$ for very general local parameters, when the ratio of weights is greater than nine. I chose to follow the approach using complex geometry in [17], rather than the algebraic approach developed in the companion manuscripts [18, 19, 20], which I have not studied in detail. The former is also closer in spirit to our discussions of Nagata’s conjecture and related topics, for example [27], during the Summer REU in Algebraic Geometry at UMass Amherst.

As I read the released manuscripts [21], I wondered whether I might be their first human reader. They seemed almost entirely unedited. While it is possible that the release served solely as a benchmark demonstrating to the investors that a large language model could solve very hard problems, I suspect that the company expected that the mathematical community, either out of curiosity or out of a sense of duty to the profession, would eventually sort out these contributions. Absorbing these arguments into mainstream mathematics and assessing their impact on our field will require a considerable editorial effort from the community. This work demands expertise and time, yet brings little of the recognition associated with original research. Like many others, I believe this approach harms our profession, especially the motivation and training of young people.

At the same time, I think these manuscripts are a stunning achievement. The model managed to connect the dots and solve the problems that many of the experts thought were out of reach. I think a wholesale re-evaluation of the mathematical corpus is currently underway, with the old underbrush of mathematical conjectures rapidly being burned in the AI fire. However, it is up to the mathematical community to figure out what mathematics will look like in the future.

I hope for closer, more respectful collaboration between frontier AI companies and the academic research community. The enormous financial resources available to these companies could support a substantial, independent foundation for research in mathematics and computer science. Such a foundation could offer research grants, graduate fellowships, and access to the best frontier models, especially to young scientists pursuing their own original research. It would give the community a role in shaping this research and sharing its benefits, rather than leaving the work to opaque in-house laboratories with little accountability for its effects on the profession. I believe this investment would also benefit the AI companies themselves in the long term. The foundation could follow the National Science Foundation’s model of peer review and provide sustained support for future generations of mathematicians and computer scientists.

Acknowledgements

Acknowledgements. I am grateful to the participants in the Reading Seminar in Algebraic Geometry and 2026 Summer REU in Algebraic Geometry at UMass Amherst for many helpful discussions. This work was supported by the National Science Foundation under grant DMS-2401387. These notes were edited and illustrated with ChatGPT Astra.

Staircase interpolation

The main result of this section is Proposition 2.4, which describes a vector space of polynomials in two variables that interpolates arbitrary values on a certain triangle of lattice points. We begin by reviewing monomial schemes defined by Young diagrams and their flat deformations. We then explain the duality between interpolation on lattice points and vanishing on monomial schemes.

Let $I \subset\mathbb{Z}_{\ge0}^{2}$ be a Young diagram; thus $I$ is closed under decreasing either coordinate. The monomials indexed by points of $\mathbb{Z}_{\ge0}^{2}$ outside $I$ span an ideal in $\mathbb{C}[w,y]$ defining the finite scheme

It is supported at the origin. Its coordinate ring has a basis consisting of $y^a w^b$, $(a,b) \in I$, so $Z_I$ has length $|I|$. For the triangle $I = T_m$, $Z_I$ is the fat point of multiplicity $m+1$, with ideal $(w,y)^{m+1}$.

We will need a closely related result. For positive integers $r$, $m$, consider the Young diagram

We will realize $Z_I$ as a flat limit of $r$ fat points of multiplicity $m$. Let $X$ be a smooth irreducible complex algebraic surface, $\mathcal{B}$ a line bundle on $X$, and $W \subset H^0(X,\mathcal{B})$ a finite-dimensional space of sections. Choose local holomorphic coordinates $(w,y)$ centered at a point of $X$ and a local frame of $\mathcal{B}$ near that point. In this chart, the monomials indexed by points outside $I$ generate the ideal

Thus $Z$ is the scheme $Z_I$ of (3) in these coordinates. Its local algebra has a basis of $y^a w^b$, $(a,b) \in I$, and hence

Let $H_F(w,y)$ be the local representative of $F \in W$ in the chosen frame. In the normalized basis $\frac{y^a w^b}{a!b!}$, restriction to $Z$ is the jet map

The collision of fat points in the following lemma is a special case of [7 §3.1, Proposition 5].

Proof. Choose distinct complex numbers $c_1,\ldots,c_r$ and put $P_s(w)=\prod_{\alpha=1}^{r}(w-sc_\alpha)$, where $s$ is a parameter. The polynomial ideal $(P_s(w),y)^m$ defines a finite flat family over the $s$-line, as illustrated in Figure 1 (PDF p. 5). Indeed, there is an isomorphism of $\mathbb{C}[s]$-modules

Since $P_s(w)^{m-a}$ is monic of degree $r(m-a)$ in $w$, polynomial division shows that the quotient on the left is free over $\mathbb{C}[s]$ with basis $y^aw^b$, $(a,b)\in I$. Its rank is $rm(m+1)/2$, proving finiteness and flatness [26 Tag 00NX]. Analytify this family and restrict the parameter $s$ to a sufficiently small disk $D$ about zero, so that its supports lie in the chosen coordinate chart. The chart identifies it with a finite flat family $\mathcal{Z}\to D$ in $X$. At $s=0$, its ideal is $(w^r,y)^m$, so the special fiber is $Z$. For $s\ne0$, the roots $sc_\alpha$ are distinct. Near $(sc_\alpha,0)$, the polynomial $P_s(w)$ is $w-sc_\alpha$ times a unit, so the local ideal is $(w-sc_\alpha,y)^m$. Thus this fiber is the disjoint union of the required $r$ fat points.

Fix a basis of $W$ and use the normalized basis $\frac{y^aw^b}{a!b!}$ of the coordinate algebra of the family. Restriction to the family is then represented by a matrix holomorphic in $s$, whose value at $s=0$ represents $J$. A nonzero minor of size rank $J$ remains nonzero for small $s\ne0$. Thus some configuration of $r$ distinct fat points has restriction rank at least rank $J$. On the space of distinct $r$-point configurations in $X$, the restriction maps vary algebraically. The same argument with minors shows that the locus where the rank is at least rank $J$ is Zariski open. It is nonempty by the construction, proving the assertion for general points.

Duality between interpolation and vanishing

Let $S\subset\mathbb{Z}^2$ be a finite set of lattice points, and let $I$ be a nonempty finite Young diagram. In Lemma 1.1, $S$ indexes evaluation points and $I$ indexes the monomials used for interpolation. We now use $S$ to index Laurent monomials and $I$ to specify vanishing along a monomial scheme supported at a single point. On the algebraic torus $(\mathbb{C}^*)^2$ with coordinates $(\xi,\eta)$, put $V_S=\operatorname{Span}_{\mathbb{C}}\{\eta^j\xi^k:(j,k)\in S\}$. Set $w=\xi-1$ and $y=\eta-1$, and let $Z_I$ be the monomial scheme of (3) in these coordinates, supported at $(1,1)$. Every element of $V_S$ is regular near $(1,1)$, even when some exponents are negative, so restriction to $Z_I$ is defined.

Proof. Use the binomial polynomials $\binom{u}{a}\binom{v}{b}$, $(a,b)\in I$, in place of the monomials $u^av^b$. This is a triangular change of basis: each polynomial has leading term $u^av^b/(a!b!)$, and all its other monomials are indexed by $I$ because $I$ is a Young diagram. The evaluation matrix at points of $S$ in this basis has entries $\binom{j}{a}\binom{k}{b}$ for $(j,k)\in S$ and $(a,b)\in I$, including when $j$ or $k$ is negative. These are exactly the coefficients of $y^aw^b$ in $\eta^j\xi^k=(1+y)^j(1+w)^k$. Thus the restriction matrix in the monomial basis of the local algebra of $Z_I$ is the transpose of the evaluation matrix at points of $S$. Consequently, evaluation is surjective if and only if restriction is injective.

For ordinary fat points, this duality between jets at a torus point and interpolation on lattice points appears in Dumnicki [5 Proposition 13], González–Karu [10 Section 2], Castravet [2 Lemmas 8.2–8.4], and He [13]. Lemma 2.3 gives the same argument for any Young diagram.

Restriction in holomorphic charts. Let $X$ be a smooth irreducible complex algebraic surface, with local holomorphic coordinates $(\xi,\eta)$ taking the value $(1,1)$ at a chosen point. Put $w=\log\xi$ and $y=\log\eta$, choosing branches that vanish at $(1,1)$. Each generator $(\eta-1)^a(\xi-1)^b$ is equal to $y^a w^b$ multiplied by a holomorphic unit, so the monomial ideal of $Z_I$ is unchanged in the analytic local ring. In the normalized basis $y^a w^b/(a!b!)$, restriction of the germs $\eta^j\xi^k=e^{kw+jy}$ has the matrix

The matrix $B_0$ is the transpose of the polynomial evaluation matrix on $S$. The same calculation applies to sections of a holomorphic line bundle on $X$ expressed in a local frame. Suppose bases indexed by $S$ have local representatives converging to $e^{kw+jy}$, together with the derivatives indexed by the set $I$. Their restriction matrices then converge to $B_0$. If $B_0$ has full column rank, one of its $|S|\times|S|$ minors is nonzero and remains nonzero under sufficiently small perturbations. The nearby restriction maps are therefore injective. In § 4, we apply this observation to bases constructed from theta functions on elliptic curves degenerating to nodal curves.

Weighted monomial ideals. Let $p,q,K$ be positive integers. Put

The monomials indexed by $I^{p,q}_K$ form a basis of the quotient algebra $\mathbb{C}[w,y]/\mathcal{J}^{p,q}_K$. The points in column $t$ of $I^{p,q}_K$ have integer second coordinates satisfying $0\le b<(K-qt)/p$, so the column has height

For $(p,q)=(1,r)$ and $K=rm$, we recover the ideal of the collision of fat points in Lemma 2.2:

The triangle $S_h(\alpha)$. Let $h$ be a positive integer and $\alpha$ a positive rational number. Write $S_h(\alpha)$ for the set of lattice points in the triangle with vertices $(h,0)$, $(0,\alpha h)$, and $(0,(\alpha+9)h)$, with the upper sloping edge omitted except for the vertex $(h,0)$:

For $1\le e\le h$, the column $j=h-e$ contains exactly $9e$ points: the half-open interval $[\alpha e,(\alpha+9)e)$ has integer length $9e$ and hence contains $9e$ integers. Including the vertex $(h,0)$, we obtain

Proof. Put $\sigma= K/(ph)$, so that $\alpha= \sigma- 9$. The hypotheses give

Put $S = S_h(\alpha)$ and $N_k = \#\{j : (j,k) \in S\}$. List the nonzero row lengths $N_k$ of $S$ in decreasing order to form a Young diagram. The column $t$ of this diagram has length $\#\{k : N_k \ge t+1\}$. Thus this diagram is contained in $I_K^{p,q}$ precisely when

where $A_{t-1}$ is the height of the $(t-1)$-st column of $I_K^{p,q}$ given by (8). Under this condition, Lemma 1.1 gives interpolation of arbitrary values on $S$ by polynomials in the span of $u^a v^b$, $(a,b) \in I_K^{p,q}$. By (6), the restriction matrix $B_0$ with $I = I_K^{p,q}$ therefore has full column rank.

It remains to verify (13). For $0 < k < \sigma h$, the horizontal slice of the triangle $S_h(\alpha)$ is

whose width is

The graph of the width $w(k)$ has a peak at $(\alpha h,9h/\sigma)$; see Figure 2 (PDF p. 8).

For $t \ge2$, a slice containing at least $t$ lattice points has width strictly greater than $t-1$, since its right endpoint is excluded. Thus every height with $N_k \ge t$ lies in the open interval

This interval is illustrated in Figure 2 (PDF p. 8). When nonempty, it has length

by (12). An open interval of positive length $L$ contains at most $\lceil L\rceil$ integers, so there are at most $A_{t-1}$ such heights by (8). If the interval is empty, the bound is automatic. This proves (13) for every column of positive index. For $t=1$, all occupied heights, including the exceptional height zero, belong to $[0,\sigma h)=[0,K/p)$, which contains $\lceil K/p\rceil=A_0$ integers. This completes the proof. □

The argument allows equality in the second inequality of (12). Thus the proposition includes the case $K=3h\sqrt{pq}$ whenever this number is an integer.

Example 1 (Sixteen points). Take $(p,q)=(1,16)$, $h=4$, and $K=48=3h\sqrt{pq}$. Then $\alpha=3$. As shown in Figure 3, $S_4(3)$ has 91 points, while $I_{48}^{1,16}$ has 96 points. Proposition 2.4 says that the resulting $96\times91$ matrix has rank 91. After the transfer to the plane in § 5, this rules out a curve of degree 12 passing through 16 general points with multiplicity at least three. This is the equality case $12=3\sqrt{16}$ in Theorem A. Example 3 gives the corresponding theta series.

Example 2 (Ten points). Take $(p,q)=(1,10)$, $h=3$, and $K=30$. Here $3h\sqrt{pq}=9\sqrt{10}<30$, and $\alpha=1$. Figure 4 shows the 55 points of $S_3(1)$ and the 60 monomials indexed by $I_{30}^{1,10}$. Proposition 2.4 gives a $60\times55$ matrix of rank 55. Since $J_{30}^{1,10}=(w^{10},y)^3$, this is the interpolation calculation for ten points of multiplicity three. Example 4 explains how it rules out plane curves of degree 9 through 10 general triple points. Example 5 extends these conditions to arbitrary multiplicity at one point.

Interpolation on the normal ruled surface

Let $C \subset\mathbb{P}^{2}$ be a smooth plane cubic, with normal bundle $M = N_{C/\mathbb{P}^{2}} = \mathcal{O}_{C}(3)$ of degree nine. Let $\operatorname{Tot}(M)$ be its total space. Its projective completion is a ruled surface

We use the convention that projective bundles parametrize lines rather than quotients. The summands $\mathcal{O}_{C}$ and $M$ of $\mathcal{O}_{C} \oplus M$ define the zero section $C_{0}$ and the infinity section $C_{\infty}$ of $\bar{\Sigma}$, respectively, so $\bar{\Sigma} \setminus C_{\infty} = \operatorname{Tot}(M)$. The projective bundle formulas give

See [12, II, Proposition 7.11; V, Propositions 2.3 and 2.9].

In the next sections, we will show that the specialization $p = 1$, $q = r$, $K = rm$ gives Theorem A (Nagata Conjecture) and that the general case gives Theorem C of § 6 about the weighted blow-up of $\mathbb{P}^{2}$ with weights $(p,q)$ at one point for very general local parameters.

Strategy of the proof. Let $v$ denote the tautological section of $\pi^{*}M$ on $\operatorname{Tot}(M)$, which becomes a vertical fiber coordinate in a local frame of $M$ on $C$. On $\operatorname{Tot}(M)$, we have $\mathcal{O}(hC_{0}) \cong\pi^{*}M^{h}$ by (14). We have

More precisely, every global section of $\pi^{*}M^{h}$ on $\operatorname{Tot}(M)$ can be written uniquely as $\sum_{j=0}^{\infty}(\pi^{*}g_{j})v^{j}$, where $g_{j} \in H^{0}(C,M^{h-j})$. On $\bar{\Sigma}$, we have $\mathcal{O}(hC_{0}) \cong(\pi^{*}M^{h})(hC_{\infty})$ by (14). Thus every global section of $\mathcal{O}(hC_{0})$ on $\bar{\Sigma}$ restricts to $\operatorname{Tot}(M)$ as a unique sum

The summation only goes up to $h$ because the order of the poles along $C_{\infty}$ is bounded from above by $h$. The Riemann–Roch theorem gives $h^{0}(C,M^{h-j}) = 9(h-j)$ for $j < h$, while $h^{0}(C,\mathcal{O}_{C}) = 1$ [12 IV, Theorem 1.3]. Hence

This is the cardinality of the set $S_{h}(\alpha)$ by (11). For a carefully chosen family of cubic curves degenerating to a nodal cubic, we will construct a point $x \in\operatorname{Tot}(M)$, local holomorphic coordinates $(w,y)$ centered at $x$, and a basis of $H^0(\overline{\Sigma},O(hC_0))$. In the notation of (16), the basis elements will be of the form $(\pi^*g_k)v^j$ with $g_k\in H^0(C,M^{h-j})$ indexed by points $(j,k)\in S_h(\alpha)$. The local expressions of the basis elements at the point $x$ will converge to $e^{kw+jy}$ as the cubic degenerates, with every partial derivative converging to the corresponding derivative of this exponential. The restriction matrices will converge to the matrix $B_0$ in (6), and Proposition 2.4 will give (15).

The GAGA theorems allow us to construct line bundles and compute global sections in the holomorphic category [23 §12, Theorems 1–3]. For $0<\tau<1$, consider the Tate curve [24 V, §1]

The additive and multiplicative quotients are isomorphic by the exponential map. We denote the quotient map $\mathbb{C}^*\to C_\tau$ by $z\mapsto[z]$. The annulus $\tau\leq\lvert z\rvert\leq1$ is a fundamental domain in $\mathbb{C}^*$, with its outer boundary identified with its inner boundary by the dilation $z\mapsto\tau z$.

For $d\in\mathbb{Z}$ and $\gamma\in\mathbb{C}^*$, let $B_{d,\gamma}$ be the line bundle on $C_\tau$ obtained from the trivial bundle $\mathbb{C}^*\times\mathbb{C}$ on $\mathbb{C}^*$ by the $\mathbb{Z}$-action generated by $(z,t)\mapsto(\tau z,\gamma z^{-d}t)$. In the covering frame, a holomorphic section of $B_{d,\gamma}$ lifts uniquely to a holomorphic function $f:\mathbb{C}^*\to\mathbb{C}$ satisfying

Functions of this form are called *multiplicative theta functions of order $d$. Conversely, every such function descends to a section. Multiplication of the covering frames gives

The multiplicative theta function $\Theta_\tau(z)$ is defined as the infinite product

which converges normally on compact annuli, has simple zeros exactly at $\tau^{\mathbb{Z}}$, and satisfies

as described in [22 §4.1.2]. Thus $\Theta_\tau(-z/\gamma)$ lifts a section of $B_{1,\gamma}$ with divisor $[-\gamma]\in C_\tau$. In particular, $\deg B_{1,\gamma}=1$, and the tensor product rule gives $\deg B_{d,\gamma}=d$ for every integer $d$.

Proof. Comparing coefficients in the Laurent series for $f(\tau z)=\gamma z^{-d}f(z)$ gives the recurrence (18). Thus one coefficient in each residue class modulo $d$ determines all the others. These $d$ coefficients can be chosen arbitrarily. Indeed, on a compact annulus, the terms of each series (19) are bounded by $c_1\exp(-c_2s^2+c_3\lvert s\rvert)$ for constants $c_1,c_2>0$ and $c_3$, since $d>0$ and $\log\tau<0$. This bound is summable over $s\in\mathbb{Z}$: for large $\lvert s\rvert$, it is at most $c_1\exp(-c_2s^2/2)$. The sum is holomorphic [25 Chapter 2, Theorem 5.2], and reindexing the series gives $f(\tau z)=\gamma z^{-d}f(z)$. Thus $f(z)$ descends to a global section of $B_{d,\gamma}$. This proves the basis claim.

As $\tau\to0$, the Tate curve $C_\tau$ degenerates to a nodal curve whose normalization is $\mathbb{P}^1$ with $0$ and $\infty$ identified [11 §5.2, Proposition 5.7 and Exercise 47]. To motivate the subsequent choices, we recall that the nodal degeneration and its vanishing cycle can be seen in the plumbing chart

Here we fix $\varepsilon$ such that $0<\varepsilon<1$ and $0<\tau<\varepsilon^2$. These inequalities guarantee that the plumbing chart is contained in the fundamental annulus. The latter is diffeomorphic to the cylinder

The angular circle $\gamma_\tau(\theta)=[\sqrt{\tau}e^{i\theta}]$ in $C_\tau$ has coordinates $(X,Y)=(\sqrt{\tau}e^{i\theta},\sqrt{\tau}e^{-i\theta})$ and shrinks into the node $(0,0)$ as $\tau\to0$. It is the vanishing cycle in Figure 5.

A fixed small $w$-disk centered at the origin gives the moving chart $z=\sqrt{\tau}e^w$ of $C_\tau$ centered at $z_0=\sqrt{\tau}$ for all sufficiently small $\tau$. This point approaches the node as $\tau\to0$. We now normalize the theta basis of Lemma 3.1 in the moving chart.

Proof. The interval $(U,U+d)$ contains one integer in each residue class modulo $d$. In Lemma 3.1, take $\gamma=\tau^{U+d/2}$ and normalize the theta basis by setting

Substituting $z=\sqrt{\tau}e^w$ gives the normalized theta series (20).

The term with $s=0$ is $e^{kw}$. For $s \ge1$, the powers of $\tau$ in the terms $s$ and $-s$ are

respectively. Both are positive. On the disc $|w| \le R$, the sum of the remaining terms is therefore bounded by

These are geometric series once $\tau$ is small. To control derivatives, write $g_\tau(w)=\widehat{\vartheta}_{k,\tau}(\sqrt{\tau}e^w)-e^{kw}$. For $R,\rho>0$, apply the same bound on $|w|\le R+\rho$. Cauchy’s estimate on a circle of radius $\rho$ about each point with $|w|\le R$ gives, for every $\ell\ge0$,

See [25 Chapter 2, Corollary 4.3 and Theorem 5.3].

Proof of Theorem B. Put $\alpha=K/(ph)-9>0$, as in Proposition 2.4. We are looking for a line bundle $M_\tau$ on $C_\tau$ of degree $9$ so that normalized global sections of $M_\tau^e$, a line bundle of degree $9e$, have asymptotics $e^{kw}$ for $k$ in the $j=h-e$ column of the triangle $S_h(\alpha)$, which requires $\alpha e\le k<(\alpha+9)e$ by (10). By Lemma 3.2, the line bundle $B_{d,\tau}^{U+d/2}$ has normalized theta functions tending to $e^{kw}$ for the integers $U<k<U+d$. Thus we need to take

where the small correction $-\delta$ is there to handle the half-open endpoints. Concretely, write $\alpha=A/D$ in lowest terms and put $\delta=\frac{1}{2Dh}$. For $1\le e\le h$, we have $M_\tau^e=B_{9e,\tau}^{9e/2+e(\alpha-\delta)}$. Lemma 3.2 applies with $U=e(\alpha-\delta)$ and degree $9e$. Since $\alpha e\in D^{-1}\mathbb{Z}$ and $0<e\delta<1/D$, we have

In particular, the endpoints on the left are nonintegral. By (10), the normalized theta functions of $M_\tau^e$ therefore tend to $e^{kw}$ for exactly the integers $k$ occurring in column $j=h-e$ of $S_h(\alpha)$.

We work in the fixed covering frame of $M_\tau$ and its tensor powers. The chart $(z,v)=(\sqrt{\tau}e^w,e^y)$ embeds a fixed polydisk $|w|,|y|<R<\pi$ into $\operatorname{Tot}M_\tau$ once $|\log\tau|>2R$. Indeed, if two points $(w,y)$ and $(w',y')$ have the same image, then $w'-w=s\log\tau+2\pi it$ for integers $s,t$. The bounds on real and imaginary parts force $s=t=0$. The map $y\mapsto e^y$ is injective on this disk as well.

Let $\overline{\Sigma}_\tau=\mathbb{P}_{C_\tau}(\mathcal{O}\oplus M_\tau)$. For each $1\le e\le h$, multiply the normalized theta basis of $H^0(C_\tau,M_\tau^e)$ by $v^{h-e}$. Together with $v^h$, these sections form a basis of $H^0(\overline{\Sigma}_\tau,\mathcal{O}(hC_0))$ by (16). The basis is indexed by $S_h(\alpha)$: the sections coming from $M_\tau^e$ correspond to $(j,k)$ with $j=h-e$ and $\alpha e\le k<(\alpha+9)e$, and $v^h$ corresponds to $(h,0)$.

In the chart $(z,v)=(\sqrt{\tau}e^w,e^y)$ of $\operatorname{Tot}(M_\tau)$, using the covering frame of $M_\tau$ and its tensor powers, the section indexed by $(j,k)$ converges locally to $e^{kw+jy}$. By Lemma 3.2 and (21), this convergence holds uniformly on compact subsets, together with every derivative in $w$. The dependence on $y$ is entirely through the factor $e^{jy}$, so differentiating $a$ times in $y$ simply multiplies the local expression by $j^a$. Thus the mixed derivative $\partial^{a+b}/(\partial y^a\partial w^b)$, evaluated at $(w,y)=(0,0)$, converges to $j^ak^b$.

The evaluation matrix therefore converges entrywise to (6), with basis sections indexed by $S_h(\alpha)$ and derivatives indexed by $I_K^{p,q}$. By Proposition 2.4, the limiting matrix has a nonzero minor of size $|S_h(\alpha)|$. This minor remains nonzero for all sufficiently small positive $\tau$, proving (15).

The line bundles $M_{\tau}$ are normal bundles of plane cubics. Indeed, the degree-three line bundle $L_{\tau}=B_{3,\tau^{3/2+(\alpha-\delta)/3}}$ satisfies $L_{\tau}^{3}=M_{\tau}$ and embeds $C_{\tau}$ as a smooth plane cubic, with normal bundle $L_{\tau}^{3}=M_{\tau}$ [12 IV, Example 3.3.3]; see also [26 Tags 0H2V and 0B3P].

Example 3 (Theta series for sixteen points). We return to Example 1 and Figure 3. Here $(p,q)=(1,16)$, $h=4$, and $K=48$, so $\alpha=3$, $\delta=1/8$, and $M_{\tau}=B_{9,\tau^{59/8}}$. We describe explicitly the basis theta function associated with each point of $S_{4}(3)$. For $1\le e\le4$, the points in column $j=4-e$ are $(4-e,k)$ for $3e\le k<12e$. The corresponding normalized theta functions, multiplied by the fiber factor $v^{4-e}$, have local expressions

The term with $t=0$ is $e^{kw+(4-e)y}$, whose exponents record the lattice point $(4-e,k)$. Every other term has a positive power of $\tau$. By Lemma 3.2, the sum of these remaining terms tends to zero uniformly on compact subsets, together with all derivatives. For example, the point $(3,3)$ corresponds to $e=1$ and $k=3$. Displaying the terms with $t=0,1,-1$, we obtain

Proof of the Nagata Conjecture

We first transfer Theorem B from the ruled surface to the plane. We fix a smooth cubic for which Theorem B holds and then degenerate the plane to its normal cone.

Proof. We use a standard deformation to the normal cone [8 §5.1]; see also [12 II, Theorem 8.24] and [26 Tag 0FBG]. See the construction in Figure 6. We blow up $C\times\{0\}$ in $\mathbb{P}^{2}\times\mathbb{A}^{1}$. Write $\mathcal{Y}$ for the resulting threefold, $s$ for the parameter on $\mathbb{A}^{1}$, $\rho:\mathcal{Y}\to\mathbb{P}^{2}$ for the projection, and $\mathcal{E}$ for the exceptional divisor. The central fiber of $\mathcal{Y}$ is $Q\cup\mathcal{E}$, where

The weighted ideal has finite colength in the holomorphic local ring of $\operatorname{Tot}(M)$. Choose $N$ such that it contains $\mathfrak{m}^{N+1}$, where $\mathfrak{m}$ is the maximal ideal of its support. Replace $(w,y)$ by algebraic regular parameters $(u,v)$ with the same $N$-jet. Their monomials agree modulo $\mathfrak{m}^{N+1}$, so they define the same weighted ideal in the holomorphic local ring of $\operatorname{Tot}(M)$.

The morphism $\mathcal{Y}\setminus Q\to\mathbb{A}^{1}$ is flat and has smooth fibers. Therefore, it is smooth. After an étale change of the parameter curve, extend the point of $\operatorname{Tot}(M)$ to a section of $\mathcal{Y}\setminus Q\to\mathbb{A}^{1}$ [26 Tag 05TG]. Lift $u,v$ to regular functions near the section and subtract their restrictions to it to make them vanish along the section. Their relative differentials are linearly independent at the original point and hence remain linearly independent along the section after shrinking. They are therefore relative local parameters along the section. The ideal $(u^{b}v^{a} : pb + qa \geq K)$ defines a finite flat scheme $Z$ supported on this section: its quotient has the fixed basis $u^{b}v^{a}$, $(a,b) \in I_{K}^{p,q}$. Its ideal sheaf is flat as well, by the exact sequence $0 \to I_Z \to\mathcal{O}_Y \to\mathcal{O}_Z \to0$.

Put $\mathcal{A} = \rho^*\mathcal{O}(3h)(-h\mathcal{E})$. Since $\mathcal{E}|_Q = C$ and $\mathcal{E} + Q$ is the divisor of $s$, we obtain

Both restrictions $\mathcal{A}|_Q$ and $\mathcal{A}|_{\mathcal{E}}$ are trivial on $C_\infty$. A global section of $\mathcal{A}$ on $Q \cup\mathcal{E}$ is a pair of sections agreeing along $C_\infty$. Its restriction to $Q$ is a constant, uniquely determined by the restriction of its $\mathcal{E}$-part to $C_\infty$. Since $Z_0$ is disjoint from $Q$, it follows that

Upper semicontinuity [12 III, Theorem 12.8] gives the same vanishing on nearby fibers. There the surface is $\mathbb{P}^2$, and the vertical twist $\mathcal{O}(-h\mathcal{E})$ restricts trivially. This proves (23).

The use of a vertical twist is related to the degeneration method of Ciliberto–Miranda [4 §2], who blow up a line in the central plane. Here the cubic gives an elliptic ruled surface.

Proof of Theorem A. First, let $d = 3h$ and suppose $9h^2 \leq rm^2$, with $r \geq10$. Apply Theorem B and Proposition 4.1 with

They give local parameters on $\mathbb{P}^2$ for which

using (9). Lemma 2.2 deforms this scheme into $r$ distinct multiplicity-$m$ fat points. Injectivity of the evaluation matrix persists, so no degree-$3h$ form vanishes to order at least $m$ at $r$ general points.

For arbitrary positive integers $d,m$ with $d \leq m\sqrt{r}$, apply this result with $h = d$ and multiplicity $3m$. If a form of degree $d$ vanished to order at least $m$ at every point, its cube would be a form of degree $3d$ of multiplicity at least $3m$, which was just ruled out.

Example 4 (Ten and sixteen points). Take $r = 10$, $m = 3$, and $d = 9$, so $h = 3$, $K = 30$, and $\alpha= 1$. The columns of $S_3(1)$ are

They contain 55 points, matching the dimension of the space of forms of degree 9. The staircase $\frac{1,10}{30}$ has column lengths 30, 20, 10, so its length is 60, the number of conditions imposed by ten triple points. It follows that no curve of degree 9 passes through ten general triple points.

Example 1 and Figure 3 give the case $r = 16$, $m = 3$, $d = 12$: here $K = rm = 48$ and $h = 4$. The argument excludes curves of degree 12 through sixteen general triple points.

Monomial valuations and weighted blow-ups

Throughout this section, $p$ and $q$ are coprime positive integers with $q > 9p$. Fix a point $x \in\mathbb{P}^2$ and regular local parameters $(u,v)$ at $x$. The corresponding monomial valuation is

For a section of $\mathcal{O}_{\mathbb{P}^2}(d)$, use any local frame at $x$. Changing the frame multiplies the local equation by a unit and does not change its valuation. Put

Thus the supremum in (26) is not attained, including when $pq$ is a square.

If one normalizes the valuation to $\nu/p$ and sets $t=q/p$, the equality (28) becomes $\widehat{\mu}(\nu/p)=\sqrt{t}$. This proves the rational-weight case with $t>9$ of the valuative Nagata conjecture of Dumnicki–Harbourne–Küronya–Roé–Szemberg [6 Conjecture 4.3]. Their stronger conjecture has the threshold $t\geq8+1/36$ [6 Conjecture 2.4]. For recent equivalent formulations in the broader setting of plane valuations, see [9 Conjecture C and Corollary 3.14]. The relation with Nagata’s conjecture through collisions of fat points already appears in [6 Proposition 4.4].

It is important to emphasize that the local parameters need to be very general. For example, in affine linear coordinates the line $v=0$ has valuation $q>\sqrt{pq}$. To specify a finite-dimensional parameter space, let $\mathfrak{m}_x$ be the maximal ideal at $x$ and put $J=\lfloor q/p\rfloor$. The pairs $(u,v)\in(\mathfrak{m}_x/\mathfrak{m}_x^{J+1})^2$ with independent linear terms form an irreducible open subset, denoted $\mathcal{J}_J(x)$. This finite jet determines the valuation in every degree. Indeed, changing the parameters modulo $\mathfrak{m}_x^{J+1}$ adds terms of valuation at least $p(J+1)>q$. The change of parameters therefore induces the identity on the associated graded ring of the weighted filtration. In particular, every valuation ideal is unchanged. We use “very general” in this fixed finite-dimensional parameter space; compare [6 Remark 2.3].

Proof of Theorem C. For each $h \ge1$, put $K_h=\left\lfloor3h\sqrt{pq}\right\rfloor$. Theorem B and Proposition 4.1 give local parameters on $\mathbb{P}^2$ for which the restriction of $H^0(\mathcal{O}(3h))$ to $\mathcal{O}/\mathcal{J}_{K_h}^{p,q}$ is injective. A projective automorphism moves the support of this finite scheme to the fixed point $x$.

For fixed $d,K$, failure of injectivity is a closed condition on $\mathcal{J}(x)$. Indeed, choose affine coordinates at $x$ and polynomial representatives of the parameter jets. The coefficients of the inverse formal change of coordinates, to any fixed order, are regular functions of these coefficients and the inverse Jacobian determinant. Expanding the degree-$d$ forms and retaining the terms indexed by $I_K^{p,q}$ gives a matrix of regular functions. Its maximal minors define the noninjectivity locus. The valuation ideals are independent of the chosen representatives.

For $(d,K)=(3h,K_h)$, the local parameters just constructed show that this closed locus is proper. Choose a jet outside the union of these loci over all $h \ge1$. If a nonzero degree-$d$ form had $\nu(F)\ge d\sqrt{pq}$, then

Since the valuation is integral, $\nu(F^3)\ge K_d$, contradicting injectivity in degree $3d$. This proves (27).

For the reverse asymptotic inequality, the length of the weighted scheme is

where $A_a=\max\left\{0,\left\lceil\frac{K-qa}{p}\right\rceil\right\}$ by (8). The estimate follows by summing $(K-qa)/p$ for $0\le a<K/q$; replacing each term by its ceiling changes the sum by $O(K)$. For $0<c<\sqrt{pq}$ and $K=\lceil cd\rceil$, the length (29) is less than $h^0(\mathbb{P}^2,\mathcal{O}(d))=\binom{d+2}{2}$ for large $d$. Hence some nonzero degree-$d$ form has valuation at least $K$ for every choice of parameters. Thus $\widehat{\mu}(\nu)\ge c$ for every $c<\sqrt{pq}$, proving the equality claimed in the theorem. □

Example 5. For the weights $(p,q)=(1,10)$ and any $m\ge1$,

Writing $F=\sum_{i\ge0}f_i(u)v^i$, we can express these conditions as $\operatorname{ord}_u f_i\ge10(m-i)$ for $0\le i<m$. Theorem C gives $d>m\sqrt{10}$ for every nonzero degree-$d$ form satisfying these conditions for very general local parameters. For $m=3$, this is the ideal in Example 2: degree 9 is excluded, whereas in degree 10, the dimension of the space of global forms is 66, while there are only 60 linear conditions. For $m=5$, degree 15 is excluded, whereas in degree 16 the dimension of the space of global forms is 153, while there are only 150 linear conditions. Thus the least possible degrees in these two cases are 10 and 16, respectively.

Example 6. Take $(p,q)=(4,49)$, so that $q/p=(7/2)^2>9$ and $\sqrt{pq}=14$. The ideal

has colength $49+37+25+13=124$. For very general parameters, Theorem C excludes a nonzero form of degree at most 14 in this ideal. Degree-fifteen forms, however, have dimension $136>124$. Hence the least degree satisfying these vanishing conditions is exactly fifteen.

In particular, no curve of degree 14 has a local equation $u^{49}-v^4$ plus terms of higher $(4,49)$-weighted degree in these prescribed very general parameters. The model cusp $u^{49}=v^4$ has normalization $u=t^4$, $v=t^{49}$. Its delta invariant is the number of gaps in the semigroup $\langle4,49\rangle$, namely $12+24+36=72$. This is smaller than the arithmetic genus $(14-1)(14-2)/2=78$ of a degree-fourteen plane curve. Thus the genus bound alone does not explain the exclusion.

The weighted blow-up. Let $\pi:Y\to\mathbb{P}^{2}$ be the normalization of the blow-up of the ideal sheaf $(u^{q},v^{p})$ (extended by the unit ideal away from $x$). This is the $(p,q)$-weighted blow-up of $\mathbb{P}^{2}$ with respect to the local parameters $(u,v)$. It is a projective surface with cyclic quotient singularities and one prime exceptional divisor $E$. Write $H$ for the pullback of the line class. We have

See, for example, [1 Theorem 4.3]. Every nonexceptional prime divisor of $Y$ is the strict transform of a plane curve $F=0$ of degree $d$, with class $dH-\nu(F)E$. Put

The class $D$ is strictly nef and has square zero. No nonzero effective $\mathbb{R}$-divisor has a class on its ray. The Cox ring of $Y$ is not finitely generated.

Proof. Theorem C and (30) give

Thus $D$ has positive intersection with every prime curve. In particular, no nonzero effective $\mathbb{R}$-divisor can have a class on its ray. A class $xH-yE$ is nef precisely when $y\geq0$ and $x\geq y/\sqrt{pq}$. This proves the first formula. Every effective divisor is a sum of $E$ and strict transforms with $\nu(F)/d<\sqrt{pq}$. The dimension count in the proof of Theorem C gives effective classes whose ratios approach $\sqrt{pq}$, proving the second formula.

The divisor class group is generated by $H$ and $E$. If the Cox ring were finitely generated, the classes of finitely many homogeneous generators would generate the closed effective cone. Each extremal ray would then contain a nonzero effective class, contradicting the assertion about $D$.