Unbounded negativity on rational
surfaces in positive characteristic

A smooth projective surface $X$ over an algebraically closed field is said to have Bounded Negativity if there exists a positive integer $b(X)$ such that $C^{2}\geq-b(X)$ for any reduced curve $C\subset X$. A folklore conjecture, going back to Enriques and discussed in [Har10, Conjecture I.2.1] and [BHK⁺13, Conjecture 1.1], is the

Bounded Negativity Conjecture. — Any smooth projective surface in characteristic $0$ has Bounded Negativity.

The assumption on the characteristic cannot be dropped: if $C$ is a curve over $\bar{\mathbf{F}}_{p}$, then the graph $\Gamma_{F^{e}}\subseteq C\times C$ of the $p^{e}$-th power Frobenius endomorphism has self-intersection $p^{e}(2-2g)$, which becomes arbitrarily negative as $e\to\infty$ when $g\geq 2$. Nonetheless, it is conceivable that certain geometric assumptions on the surface may still guarantee Bounded Negativity in positive characteristic. For instance, [BBC⁺12, discussion preceding Example 3.3.3] and [Har18, Conjecture 2.1.2] ask whether smooth rational surfaces over a field of positive characteristic have Bounded Negativity. We give a negative answer to this question:

Main Theorem. — Let $k$ be an algebraically closed field of characteristic $p>0$, let $m$ be a positive integer invertible in $k$, and let $R_{m}$ be the blowup of $\mathbf{P}^{2}$ along

Let $C_{1}=V(x_{0}+x_{1}+x_{2})\subseteq\mathbf{P}^{2}$, and for $d\geq 1$ invertible in $k$, write $C_{d}\subseteq\mathbf{P}^{2}$ for the image of

If $dm=p^{e}-1$ for some positive integer $e$, then the strict transform $\widetilde{C}_{d}\subseteq R_{m}$ of $C_{d}$ is a smooth rational curve with $\widetilde{C}_{d}^{2}=d(3-m)-1$. Thus, if $m>3$, the rational surface $R_{m}$ does not have Bounded Negativity over $k$.

Since $\mathbf{P}^{2}$ has Bounded Negativity, this shows that [BDRH⁺15, Problem 1.2] has a negative answer in positive characteristic:

Corollary. — Bounded Negativity is not a birational property of smooth projective surfaces in positive characteristic.   $\blacksquare$

In fact, since every smooth projective surface $X$ admits a finite morphism $X\to\mathbf{P}^{2}$, pulling back the blowup $R_{m}\to\mathbf{P}^{2}$ gives a blowup $\widetilde{X}\to X$ with a finite morphism $\widetilde{X}\to R_{m}$. Pulling back the curves $\widetilde{C}_{d}$ to $\widetilde{X}$ shows:

Corollary. — If $X$ is a smooth projective surface over an algebraically closed field $k$ of positive characteristic, then there exists a blowup $\widetilde{X}\to X$ such that $\widetilde{X}$ does not have Bounded Negativity.   $\blacksquare$

In §​​ 1, we give a direct proof of the Main Theorem. In §​​ 2, we realise the plane curves $C_{d}$ as norms of line configuration, thereby deriving equations for them. In §​​ 3, we view $R_{m}$ as an isotrivial family of diagonal curves over $C_{1}$ and relate the curves $\widetilde{C}_{d}$ on $R_{m}$ to graphs of Frobenius morphisms on Fermat curves. Finally, we close in §​​ 4 with some questions and remarks towards characteristic zero.

Sections 2 and 3 each give alternative methods for computing the self-intersections of $\widetilde{C}_{d}$. Given the simplicity of the formulas for $\widetilde{C}_{d}$ and the many connections to other well-studied examples, it is surprising that these curves have not been found before.

Throughout the paper, $k$ will be an algebraically closed field of arbitrary characteristic, and $m$ and $d$ will denote positive integers invertible in $k$. We will use the notation of the Main Theorem throughout.

In this section, fix $m$ and write $R\coloneqq R_{m}$ for the blowup of $\mathbf{P}^{2}$ along $Z\coloneqq Z_{m}$. The generators $s_{0}=x_{1}^{m}-x_{2}^{m}$, $s_{1}=x_{2}^{m}-x_{0}^{m}$, and $s_{2}=x_{0}^{m}-x_{1}^{m}$ of the ideal of $Z$ give a closed immersion $R\hookrightarrow\mathbf{P}^{2}\times\mathbf{P}^{2}$. Since $s_{0}+s_{1}+s_{2}=0$, one of the $s_{i}$ can be eliminated at the expense of breaking the symmetry in the computations below.

1.1. Lemma. — The embedding $R\hookrightarrow\mathbf{P}^{2}\times\mathbf{P}^{2}$ realises $R$ as the complete intersection

of degrees $(0,1)$ and $(m,1)$ in $\mathbf{P}^{2}\times\mathbf{P}^{2}$. In particular, $K_{R}=\mathcal{O}_{R}(m-3,-1)$.

Proof. The generators $s_{0},s_{1},s_{2}$ of the ideal of $Z$ identify $R$ as

The relation $s_{0}+s_{1}+s_{2}=0$ shows that $R$ is contained in the locus $y_{0}+y_{1}+y_{2}=0$. The equation $y_{0}(x_{2}^{m}-x_{0}^{m})=y_{1}(x_{1}^{m}-x_{2}^{m})$ can be rewritten as $(y_{0}+y_{1})x_{2}^{m}=x_{0}^{m}y_{0}+x_{1}^{m}y_{1}$, which is equivalent to $x_{0}^{m}y_{0}+x_{1}^{m}y_{1}+x_{2}^{m}y_{2}=0$ since $y_{0}+y_{1}+y_{2}=0$. The same holds for the other two equations by symmetry. The final statement then follows from the adjunction formula since $K_{\mathbf{P}^{2}\times\mathbf{P}^{2}}=\mathcal{O}(-3,-3)$.   $\blacksquare$

Alternatively, one can observe that the complete intersection of Lemma 1 maps birationally onto its first factor, where the fibres are points when $[x_{0}^{m}:x_{1}^{m}:x_{2}^{m}]\neq[1:1:1]$ and lines otherwise.

1.2. Lemma. — If $\operatorname{char}k=p>0$ and $dm=p^{e}-1$ for some positive integer $e$, then the map $\widetilde{\phi}_{d}\colon C_{1}\to\mathbf{P}^{2}\times\mathbf{P}^{2}$ given by

lands in $R$. In particular, it is the unique map lifting $\phi_{d}\colon C_{1}\to\mathbf{P}^{2}$.

Proof. Since $x_{0}+x_{1}+x_{2}=0$, the image of $\widetilde{\phi}_{d}$ is contained in the locus $y_{0}+y_{1}+y_{2}=0$. Since $dm=p^{e}-1$, the expression $x_{0}^{m}y_{0}+x_{1}^{m}y_{1}+x_{2}^{m}y_{2}$ pulls back to $x_{0}^{p^{e}}+x_{1}^{p^{e}}+x_{2}^{p^{e}}$, which vanishes because the $p^{e}$-th power Frobenius is an endomorphism. Thus $\widetilde{\phi}_{d}$ is a lift of $\phi_{d}$ to $R$, and it is the unique lift since the first projection $\operatorname{pr}_{1}\colon R\to\mathbf{P}^{2}$ is birational.   $\blacksquare$

1.3. Corollary. — The map $\widetilde{\phi}_{d}\colon C_{1}\to R$ is a closed immersion, whose image $\widetilde{C}_{d}$ is a smooth rational curve in $R$ with $\widetilde{C}_{d}^{2}=d(3-m)-1$.

Proof. The first two statements follow from the coordinate expression in Lemma 1, since $\widetilde{\phi}_{d}$ embeds $C_{1}$ linearly into the second factor. The same expression shows that $\widetilde{\phi}_{d}^{*}\mathcal{O}_{R}(a,b)=\mathcal{O}_{C_{1}}(da+b)$, so $K_{R}\cdot\widetilde{C}_{d}=d(m-3)-1$ by Lemma 1. Then the adjunction formula shows that

This completes the proof of the Main Theorem.   $\blacksquare$

A consequence of Corollary 1 is that the singularities of $C_{d}$ are contained in $Z$. However, the individual multiplicities are not so easy to determine. For example, in Lemma 3 we will compute the multiplicity of $C_{d}$ at $[1:1:1]$ in terms of point counts on Fermat curves.

In this section, we observe that the $d$-th power maps $\pi_{d}\colon\mathbf{P}^{2}\to\mathbf{P}^{2}$ are finite Galois morphisms such that $\pi_{d}^{*}C_{d}$ is the union of the Galois translates of $C_{1}$. Thus the $C_{d}$ are norms of line configurations, from which we derive in Corollary 2 a formal product formula for the equation of the plane curves $C_{d}$. In the second half of this section, we observe that in characteristic $p>0$ and for $q$ a power of $p$, the curve $C_{q-1}$ comes from a subconfiguration of the set of all $\mathbf{F}_{q}$-rational lines. This allows us to show in Corollary 2 that an equation of $C_{q-1}$ in this case is the complete homogeneous polynomial of degree $q-1$.

2.1. Power Maps. For any integer $a\geq 1$ invertible in $k$, write $\pi_{a}$ for the $a$-th power map $\mathbf{P}^{2}\to\mathbf{P}^{2}$. Since $\pi_{a}^{*}Z_{m}=Z_{am}$, the map $\pi_{a}$ lifts to a finite morphism $\widetilde{\pi}_{a}\colon R_{am}\to R_{m}$ given by

Since $a$ is invertible in $k$, both $\pi_{a}$ and $\widetilde{\pi}_{a}$ are finite Galois with group $G=\boldsymbol{\mu}_{a}^{3}/\boldsymbol{\mu}_{a}$, where $(\zeta_{0},\zeta_{1},\zeta_{2})\in G$ acts on $\mathbf{P}^{2}$ via

This gives a tower of extensions

indexed by the poset of positive integers invertible in $k$ under the divisibility relation.

2.2. Lemma. — If $a,d\geq 1$ are invertible in $k$, then

1. (i)

   The map $\phi_{d}\colon C_{1}\to\mathbf{P}^{2}$ is unramified and birational onto its image;

2. (ii)

   The inverse image $\pi_{a}^{*}C_{ad}$ is totally split into the $G$-translates of $C_{d}$.

Proof. The Jacobian $(d\cdot x_{0}^{d-1},d\cdot x_{1}^{d-1},d\cdot x_{2}^{d-1})$ of $\phi_{d}$ only vanishes when $x_{0}=x_{1}=x_{2}=0$, showing that $\phi_{d}$ is unramified. Then the map $C_{1}\to C_{d}^{\nu}$ to the normalisation of $C_{d}$ is unramified, hence an isomorphism since it is an étale map of smooth projective rational curves, proving (i).

Since $\pi_{a}\circ\phi_{d}=\phi_{ad}$, part (i) shows that $\pi_{a}$ maps $C_{d}$ birationally onto its image. This shows that the decomposition group of $C_{d}$ is trivial, so no two $G$-translates $\zeta C_{d}$ of $C_{d}$ coincide and $C_{ad}$ is totally split under $\pi_{a}$.   $\blacksquare$

2.3. Corollary. — If $d$ is invertible in $k$, then the homogeneous ideal of $C_{d}\subseteq\mathbf{P}^{2}$ is generated by

Proof. By Lemma 2 (ii), the inverse image $\pi_{d}^{-1}(C_{d})$ is the union of lines $\bigcup_{\zeta\in\boldsymbol{\mu}_{d}^{3}/\boldsymbol{\mu}_{d}}\zeta C_{1}$. The result follows since $C_{1}$ is cut out by $x_{0}+x_{1}+x_{2}=0$.   $\blacksquare$

2.4. In general, the $f_{d}$ are complicated symmetric polynomials. However, in Corollary 2 we will show that the coefficients of $f_{q-1}$ are congruent to $1$ modulo $p$ if $q$ is a power of a prime $p$. For example, for $q=p=3$, we get

In the remainder of this section, assume $\operatorname{char}k=p>0$ and let $q$ be a power of $p$.

2.5. Finite Field Line Configurations. The configuration of $\mathbf{F}_{q}$-rational lines in $\mathbf{P}^{2}$ is the union of the lines $L_{a}=\{a_{0}x_{0}+a_{1}x_{1}+a_{2}x_{2}=0\}$ indexed by $a=[a_{0}:a_{1}:a_{2}]\in\check{\mathbf{P}}^{2}(\mathbf{F}_{q})$. Their union is the divisor in $\mathbf{P}^{2}$ cut out by the polynomial

since the three columns become linearly dependent when $x_{0}$, $x_{1}$, and $x_{2}$ satisfy a linear relation over $\mathbf{F}_{q}$, and the degree equals $q^{2}+q+1=|\check{\mathbf{P}}^{2}(\mathbf{F}_{q})|$. Now Lemma 2(ii) shows that $\pi_{q-1}^{*}C_{q-1}$ consists of the $q^{2}-2q+1$ lines $L_{a}$ with all coordinates of $a=[a_{0}:a_{1}:a_{2}]$ nonzero. We can thus derive an equation for $C_{q-1}$ by extracting factors cutting out the lines $L_{a}$ in which $a$ has a vanishing coordinate. A neat description of the result comes from the following polynomial identity, also observed in [RVVZ01, p. 90]:

2.6. Lemma. — For any nonnegative integer $n$, define the polynomials

in $\mathbf{Z}[x_{0},x_{1},x_{2}]$. Then $h_{2}=(x_{2}-x_{1})(x_{0}-x_{2})(x_{1}-x_{0})$ and $h_{n}=h_{2}g_{n-2}$ for $n\geq 3$.

Proof. Let $G(t)\coloneqq\sum_{n\geq 0}g_{n}t^{n}$ and $H(t)\coloneqq\sum_{n\geq 0}h_{n}t^{n}$ be the generating functions of the $g_{n}$ and $h_{n}$, respectively. A standard computation gives

On the other hand, writing $h_{n}=(x_{2}-x_{1})x_{0}^{n}+(x_{0}-x_{2})x_{1}^{n}+(x_{1}-x_{0})x_{2}^{n}$ gives

The result follows by recognising the numerator as $h_{2}$.   $\blacksquare$

2.7. Corollary. — Suppose $\operatorname{char}k=p>0$ and $q$ is a power of $p$. Then $g_{q-1}$ generates the homogeneous ideal of $C_{q-1}\subseteq\mathbf{P}^{2}$. In particular, $f_{q-1}\equiv g_{q-1}\pmod{p}$.

Proof. Since $C_{1}=L_{[1:1:1]}$ is among the $\mathbf{F}_{q}$-rational lines of 2 and is not a coordinate axis, the points $[x_{0}:x_{1}:x_{2}]$ of $C_{1}=V(x_{0}+x_{1}+x_{2})$ satisfy the equation there divided by $x_{0}x_{1}x_{2}$:

Since $C_{1}\to C_{q-1}$ is $[x_{0}:x_{1}:x_{2}]\mapsto[x_{0}^{q-1}:x_{1}^{q-1}:x_{2}^{q-1}]$, any $[x_{0}:x_{1}:x_{2}]\in C_{q-1}$ satisfies

So $h_{q+1}$ vanishes on $C_{q-1}$, which by Lemma 2 equals $(x_{2}-x_{1})(x_{0}-x_{2})(x_{1}-x_{0})g_{q-1}$. The result follows since $C_{q-1}$ is not contained in any of the lines $\{x_{2}=x_{1}\}$, $\{x_{0}=x_{2}\}$, or $\{x_{1}=x_{0}\}$, and $\deg g_{q-1}=q-1=\deg C_{q-1}$.   $\blacksquare$

2.8. Negative Curves via Equations. If $m>3$ and $q$ is a power of $p$ congruent to $1$ modulo $m$, then the curves $\widetilde{C}_{d}\subseteq R_{m}$ with $dm=q^{e}-1$ of the Main Theorem can therefore be obtained by starting with the very explicit equations

blowing up at $[1:1:1]$, pulling back along $\widetilde{\pi}_{m}\colon R_{m}\to R_{1}$, and taking one of the $m^{2}$ isomorphic components $\zeta\widetilde{C}_{d}$ for $\zeta\in\boldsymbol{\mu}_{m}^{3}/\boldsymbol{\mu}_{m}$. From this point of view, the self-intersection may be computed as

since the intersection number between $\widetilde{C}_{d}$ and a Galois translate by $\zeta=(\zeta_{0},\zeta_{1},\zeta_{2})\in G\setminus\{1\}$ is

Indeed, $\zeta\widetilde{C}_{d}$ is the image of the morphism $\zeta\circ\widetilde{\phi}_{d}$ given by

Thus, $\widetilde{C}_{d}$ and $\zeta\widetilde{C}_{d}$ only intersect when $\zeta\widetilde{\phi}_{d}([x_{0}:x_{1}:x_{2}])=\widetilde{\phi}_{d}([x_{0}:x_{1}:x_{2}])$. At most one of the $x_{i}$ can vanish since $x_{0}+x_{1}+x_{2}=0$, so there are no intersections when $\zeta_{i}\neq\zeta_{j}$ for $i\neq j$, and a single intersection with multiplicity $d$ at $V(x_{k})$ when $\zeta_{i}=\zeta_{j}$ and $\{i,j,k\}=\{0,1,2\}$.

By Lemma 1, the second projection $\operatorname{pr}_{2}\colon R_{m}\to V(y_{0}+y_{1}+y_{2})$ realises $R_{m}$ as the family of diagonal degree $m$ curves over $C_{1}\cong\mathbf{P}^{1}$ given by

If $\operatorname{char}k=p>0$ and $m$ is invertible in $k$, then the curves $\widetilde{C}_{d}\subseteq R_{m}$ for $dm=p^{e}-1$ are given by sections $\widetilde{\phi}_{d}\colon C_{1}\to R_{m}$ of $\operatorname{pr}_{2}$. In this section, we pull back the family $R_{m}\to C_{1}$ and the sections $\widetilde{\phi}_{d}$ along finite covers of $C_{1}$. Pulling back along covers by Fermat curves allows us to relate the $\widetilde{C}_{d}$ in Corollary 3 with graphs of Frobenius on products of Fermat curves. Pulling back along the Frobenius morphism of $C_{1}$ allows us to realise the $\widetilde{C}_{d}$ in Corollary 3 as pullbacks of a constant section $\widetilde{C}_{0}$ under powers of a horizontal Frobenius morphism of $R_{m}$ over $C_{1}$.

3.1. Intermediate Surfaces. For positive integers $m$ and $n$ invertible in $k$ and $r\in\mathbf{N}$, denote by $R_{m,n,r}$ the normal surface

It is smooth if and only if $m=1$ or $r\in\{0,1\}$; in all other cases, the singular locus $V(x_{0}y_{0},x_{1}y_{1},x_{2}y_{2})$ consists of the $3n$ points

Note that $R_{m,1,1}$ is none other than the surface $R_{m}$ of Lemma 1. If $X_{n}$ denotes the Fermat curve $V(y_{0}^{n}+y_{1}^{n}+y_{2}^{n})\subseteq\mathbf{P}^{2}$ of degree $n$, then $R_{m,n,0}$ coincides with $X_{m}\times X_{n}$. The surfaces $R_{m,n,r}$ for $r>0$ come with a projection

that is smooth away from the $3n$ fibres above $V(y_{0}y_{1}y_{2})\subseteq X_{n}$, and whose singular fibres consist of $m$ lines meeting at a point.

3.2. Generalized Power Maps. For positive integers $a$ and $b$ invertible in $k$, define the finite morphism

For $b=1$ and $n=r=1$, it coincides with the morphism $\widetilde{\pi}_{a}$ from 2. When $a=1$, these fit into pullback squares

If $F^{e}\colon X_{n}\to X_{n}$ is the $p^{e}$-th power Frobenius morphism of $X_{n}$, there are pullback squares

so $R_{m,n,p^{e}r}$ is the Frobenius twist $R_{m,n,r}^{(e)}$ of $R_{m,n,r}$ over $X_{n}$. We denote the top map by $\pi^{(e)}$.

3.3. Lemma. — Let $m$ and $n$ be positive integers invertible in $k$, let $a,r\in\mathbf{Z}$, and assume that $r$ and $r+am$ are nonnegative. Then the map

maps $R_{m,n,r+am}$ birationally onto $R_{m,n,r}$.

Proof. Note that $\psi_{a}$ is a birational map with rational inverse $\psi_{-a}$. The result follows since $\psi_{a}$ takes $R_{m,n,r+am}$ into $R_{m,n,r}$ and $\psi_{-a}$ does the opposite, and neither surface is contained in the locus where $\psi_{a}$ or $\psi_{-a}$ is undefined.   $\blacksquare$

This allows us to relate $R_{m,m,m}$ and $X_{m}\times X_{m}$:

3.4. Corollary. — The surfaces $X_{m}\times X_{m}\cong R_{m,m,0}$ and $R_{m,m,m}$ are birational via

The composition $\rho\colon X_{m}\times X_{m}\dashrightarrow R_{m}$ of $\psi$ with $\pi_{1,m}$ is given by

If $\operatorname{char}k=p>0$, $m$ is invertible in $k$, and $dm=p^{e}-1$ for some positive integer $e$, then the strict transform of $\pi_{1,m}^{*}\widetilde{C}_{d}$ under $\psi$ is the transpose $\Gamma_{F^{e}}^{\top}$ of the graph of the $p^{e}$-power Frobenius.

Proof. The first statement follows by applying Lemma 3 to $m=n=r$ and $a=-1$, and the second is immediate from the definitions. For the final statement, recall that $\Gamma_{F^{e}}^{\top}$ is given by the section $s\colon X_{m}\to X_{m}\times X_{m}$ of $\operatorname{pr}_{2}$ given by

By the first pullback square of 3, the curve $\pi_{1,m}^{*}\widetilde{C}_{d}$ is the image of the section $X_{m}\to R_{m,m,m}$ given by

which agrees with $\psi\circ s$.   $\blacksquare$

3.5. The curves $\Gamma_{F^{e}}\subseteq X_{m}\times X_{m}$ are the standard example of curves with unbounded negative self-intersection: the condition $m>3$ of the Main Theorem is exactly the condition $g(X_{m})>1$ that makes $\Gamma_{F^{e}}^{2}=p^{e}(2-2g)$ negative. In fact, since $\Gamma_{F^{e}}^{\top}$ passes through $3m$ of the $3m^{2}$ points of indeterminacy of $\psi$, resolving the map shows that $m^{2}\,\widetilde{C}_{d}^{2}=\Gamma_{F^{e}}^{2}-3m$.

On the other hand, $R_{m}\to C_{1}$ is an isotrivial family of diagonal degree $m$ curves that becomes rationally trivialised over the $m$-th power cover $X_{m}\to C_{1}$. Thus, we can also look directly at the pullback $\pi^{(e)}\colon R_{m}^{(e)}\to R_{m}$ of the Frobenius $F^{e}\colon C_{1}\to C_{1}$. Note that $R_{m}^{(e)}=R_{m,1,p^{e}}$ by 3, so we get:

3.6. Corollary. — If $p^{e}=dm+1$, then $R_{m}^{(e)}$ is birational to $R_{m}$ via