Nagata's conjecture for plane curves

1. A countable exceptional set, cyclic products and normal polynomials

For a fixed degree $e$ and multiplicity vector $\mathbf n$, the locus $A_{e,\mathbf n}$ admitting a nonzero form is closed: it is defined over $\mathbf Q$ by the full-column-size minors of the Taylor-condition matrix. Remove all proper such loci to form $E_r$. A counterexample outside $E_r$ therefore has an incidence locus equal to all of $U_r$. Multiplying forms associated with the $r$ cyclic permutations gives $d=rd_0$, $m=\sum_i m_i>0$, and

with $d/m\le\sqrt r$. We write $\mathcal U_r$ for the manuscript's property $U_r(d,m)$ to distinguish it from the configuration space. The property persists under every common multiple $(Nd,Nm)$.

For a smooth degree-$k$ plane curve $B$, put $L=\mathcal O_B(1)$; its normal bundle is $M=L^k$. Move points towards $B$ and rescale the normal direction. For arbitrary distinct $a_i\in B$ and arbitrary normal vectors $c_i\in M_{a_i}$, this produces

of multiplicity at least $m$ at each $(a_i,c_i)$. Taking the lowest power of the motion parameter leaves distinct powers of $w$, ensuring nonvanishing. Multiplicity is preserved by extending the local derivative conditions, centered at the moving points, to parameter zero. The proof contradicts universal existence at a special configuration; it does not assume a reverse deformation from special to general points.

Lemma 2.1 · pp. 3–4

Proposition 2.2 · pp. 4–6

[StacksN, Tag 0B3P, Lemma 15.2, p. 26]

2. Exclude equality as well in the square case

Let $r=k^2$ with $k\ge4$. Choose $P=\sum a_i$ so that $A=M(-P)$ is nontorsion, and choose the normal-vector tuple $c$ outside the image of $H^0(B,M)\to\bigoplus_iM_{a_i}$. These choices use countability of torsion and $h^0(B,M)=(k^2+3k)/2<r$.

If $j$ is the highest exponent of $F$, then $d\le km$ implies $j\le m$. When $j<m$, the required zeros of the leading coefficient either exceed its degree or, at the boundary $d=km$, produce a nonzero section of the nontrivial degree-zero bundle $A^{m-j}$. Both are impossible. The remaining case has $j=m$, $d=km$, and a nonzero constant leading coefficient $f_m$. Each fiber restriction equals $f_m(w-c_i)^m$, so the next coefficient satisfies $(f_{m-1}(a_i))=-mf_m(c_i)$. This puts the chosen $c$ in the evaluation image, again a contradiction. This second step excludes the endpoint equality.

Proposition 2.3 · pp. 6–7

[StacksC, Tag 01XT, Lemma 8.1, p. 16]

[StacksT, Tag 0C1Z, Lemma 17.1, p. 36]

3. Nonsquare case: collision on a fixed curve, then a numerical matrix limit

The rationality of $d/m$ gives $d/m<\sqrt r$. Setting nine normal displacements to zero on a cubic forces the coefficient of degree $j<m$ to vanish along $(m-j)P$. Lemma 2.4 excludes $d\le3m$. Put $A=L^3(-P)$ and $T=L^{d-3m}$. The substitution $w=\sigma_Pv$ and division $\widetilde F=\sigma_P^{-m}F(\sigma_Pv)$ place the first $m+1$ coefficient blocks in bundles $T\otimes A^{m-j}$ of the same positive degree $h=3(d-3m)$. For $j>m$, use $\sigma_P^{j-m}H^0(B,L^{d-3j})$; their direct sum defines $W_\tau$.

Take $B=X_\tau=\mathbf C^*/\tau^{\mathbf Z}$, $0<\tau<1$, with the manuscript's theta-bundle multipliers. Degree-three sections separate points and tangent directions, giving an analytic embedding; Lemma 3.3 invokes Chow's theorem to make its image an algebraic plane cubic. The location and statement in Chow's original paper remain unchecked here.

First fix $\tau$. Collide the remaining $q=r-9$ points along $y=0$ in local coordinates $(x,y)$. After $b$ derivatives in $y$, there are zeros of order at least $m-b$ at each point. A normalized finite-dimensional subsequence has a nonzero limit; Rouché's theorem preserves the total number of zeros and gives

Next normalize the residue-class theta bases. Their local expressions converge to $e^{Kx+jy}$ together with every fixed derivative, so the matrices converge to $B_0=(K^\ell j^b)$. No continuous choice of kernel vectors across $\tau$ is used.

Lemma 2.4; Proposition 3.1 · pp. 8–9

Lemmas 3.2–3.3; §3.3 · pp. 10–12

Lemma 4.1 · pp. 13–14

Lemma 4.2; Proposition 4.3 · pp. 14–17

4. Row interpolation yields column independence and excludes every counterexample

The finite exponent set $S=\{(j,K)\}$ is defined in (4.4), including the degree-zero block $(d/3,0)$ when $3\mid d$. Put $a=3(\sqrt r-3)$, $\rho=3(d/m-3)$, and choose positive rational $\Delta$ with $\rho+\Delta\rho/9<a$. Taking $\lambda<1$ sufficiently close to one places $S$ in a contracted polygon $mD_\lambda+(0,U_0)$. Its maximum horizontal width is $m\lambda$, and the interval of heights with width at least $s$ has length $q(m\lambda-s)$. The identity $2a+a^2/9=q$ is decisive.

First take a common multiple of $(d,m)$ so that $q(1-\lambda)m\ge1$. This absorbs the lattice endpoint error and gives

Interpolate the largest rows first, multiplying each correction by $\prod_{a<i}(K-K_a)$ to preserve earlier rows. Every correction still satisfies $\ell<q(m-b)$. Hence the allowed span of $K^\ell J^b$ maps onto all functions on $S$, and $B_0$ has full column rank. A nonzero maximal minor persists for sufficiently small positive $\tau$, making $J_\tau$ injective and contradicting its nonzero kernel.

The common multiple leaves $d/m$, $\Delta$ and $\lambda$ unchanged and is taken once, before both limits. Exclusion of $\mathcal U_r(d,m)$ in both square and nonsquare cases returns the strict inequality for every degree and every unequal multiplicity vector outside the original single $E_r$.

Lemmas 5.1–5.4 · pp. 18–21

Proposition 5.5; Theorem 1.1 proof · p. 21

1. Proper incidence loci and a nonempty complement

For every violating $(d,\mathbf m)$, the incidence locus is defined by rational minors. Properness over $\mathbf C$ implies properness over $\mathbf Q$, and a nonzero rational minor stays nonzero under characteristic-zero field extension. Removing these countably many loci yields all inequalities simultaneously. Uncountability of $k$ permits $2r$ coordinates algebraically independent over $\mathbf Q$; they avoid every excluded locus and every diagonal. No embedding $k\hookrightarrow\mathbf C$ is required.

Corollary 6.1 · p. 22

1. Intersections with every irreducible curve and the self-intersection

An irreducible curve on the blowup is exceptional or the strict transform of a plane curve. In the former case $N_r\cdot E_i=1$; in the latter, $N_r\cdot\widetilde C=d\sqrt r-\sum_i\operatorname{mult}_{p_i}C>0$. Thus $N_r$ is strictly nef. Separately, $H^2=1$, $H\cdot E_i=0$ and $E_i\cdot E_j=-\delta_{ij}$ give $N_r^2=r-r=0$. Nefness is not inferred from this numerical identity alone.

Corollary 6.2 · p. 22

2. The normalized lower bound and the interpolation-count upper bound

Nefness of $N_r/\sqrt r=H-r^{-1/2}\sum_iE_i$ gives $\varepsilon\ge r^{-1/2}$. For the reverse bound, set $e_n=\lceil\sqrt r(n+1)\rceil$. Since $\binom{e_n+2}{2}>r\binom{n+1}{2}$, there is a nonzero degree-$e_n$ form $D$ of multiplicity at least $n$ at every point. If $H-t\sum_iE_i$ is nef, then

so $t\le e_n/(rn)\to1/\sqrt r$. This upper bound follows from the condition count for any configuration; the difficulty at very general points is the matching lower bound. Positive intersection with each curve is compatible with an asymptotic threshold equal to the boundary value.

Corollary 6.2 · p. 22