Horikawa surfaces

[2018-02-26 Mon]

Mostly for my own convenience, here is an overview of the geometry of Horikawa surfaces (because I will forget most of this).

Horikawa's work

A Horikawa surface is a minimal surface of general type with geometric genus $$p_g\geq 3$$ which lies on the Noether line $$c_1^2=2p_g-4$$. These all have irregularity zero. This means that the Chern numbers are given in terms of $$p_g$$ by $c_1^2=2p_g-4,\quad c_2=10p_g+16$ These surfaces were studied extensively by Horikawa. Here is a summary of what he proved. Let $$n=p_g-1$$.

• The canonical map $$\Phi_K\colon X\to\mathbf{CP}^n$$ has image a surface $$W=\Phi_K(X)$$ of degree $$n-1$$ in $$\mathbf{CP}^n$$ (Lemma 1.1). Write $$f\colon X\to W$$ for the corestriction; then $$f$$ is a double branched cover. The ramification locus $$R\subset X$$ of $$f$$ satisfies $$K_X=f^*K_W+R$$; the branch locus $$B\subset W$$ is the pushforward $$f_*R$$ and has no multiple component.
• The image $$W$$ of $$f$$ is one of the following (Lemma 1.2):
• "Type (\infty)": ($$n=2$$) the whole of $$\mathbf{CP}^2$$ or ($$n=5$$) the quadratic Veronese surface $$\mathbf{CP}^2\subset\mathbf{CP}^5$$.
• "Type (d)": ($$n\geq 3$$) a Hirzebruch surface $$\mathbf{F}_d$$ where $$d+3\leq n$$ and $$n-d-3$$ is even, embedded into $$\mathbf{CP}^n$$ via the linear system $$\left|\Delta_0+\frac{n-1+d}{2}\Gamma\right|$$ where $$\Gamma$$ is a fibre of $$\mathbf{F}_d\to\mathbf{CP}^1$$ and $$\Delta_0$$ is the unique section with square $$-d$$. In fact, we will see below that $$n\geq\max(d+3,2d-3)$$.
• "Type (d')": ($$n\geq 3$$) the cone $$C_{n-1}$$ on a rational curve of degree $$d=n-1$$ in $$\mathbf{CP}^{n-1}\subset\mathbf{CP}^n$$. In fact, this is only possible for $$n=3,4,5$$.
We will analyse the possible branch curves case-by-case below.
• If $$K^2$$ is not divisible by 8 ($$p_g\neq 2\mod 4$$) then any two Horikawa surfaces with this value of $$K^2$$ can be connected through deformations (i.e. there is a path in the Gieseker moduli space connecting these surfaces: note that this path may connect different irreducible components of the moduli space, but all surfaces along the path are diffeomorphic).
• If $$k\geq 2$$, $$K^2=8k$$ ($$p_g=4k+2$$) then there are two deformation classes of Horikawa surfaces with this value of $$K^2$$ (Theorem 7.1). These comprise, on the one hand, surfaces of type (d) with $$d$$ even and $$d\leq 2k$$, and, on the other, surfaces of type (2k+2).
• If, moreover, $$k$$ is even, then these surfaces are homotopy equivalent.
• If, instead, $$k$$ is odd, then they are distinguished by the second Stiefel-Whitney class (type $$(2k+2)$$ is spin/has even intersection form, the others are not spin/have odd intersection form).
• Finally (Theorem 4.1), if $$K^2=8$$, then there are two deformation classes: surfaces of type (0) and (2) and surfaces of type (\infty) and (4'). Surfaces from different deformation classes are not homotopy equivalent, distinguished by their second Stiefel-Whitney class.

Type ($$\infty$$)

($$n=2$$) The surface $$W$$ is the whole of $$\mathbf{CP}^2$$; the ramification locus is then homologous to $$4f^*H$$ (as $$K_X=f^*H$$ and $$K_W=-3H$$) so the branch locus $$B$$ is a curve of degree 8 in $$\mathbf{CP}^2$$.

($$n=5$$) The surface $$W$$ is the quadratic Veronese surface $$\mathbf{CP}^2\subset\mathbf{CP}^5$$. We have $$K_X=\Phi_K^*H$$, but $$\Phi_K$$ is the composition of $$f$$ with the inclusion of the (degree 2) Veronese surface, so if $$h$$ is the hyperplane class on $$\mathbf{CP}^2$$ we get $$\Phi_K^*H=2f^*h$$. We also have $$f^*K_W=-3g^*h$$, so $$R=5g^*h$$ and the branch locus $$B$$ is a curve of degree 10.

Type ($$d$$)

($$n\geq 3$$) The surface $$W$$ is a Hirzebruch surface $$\mathbf{F}_d$$ where $$n-d-3$$ is a nonnegative even integer, embedded into $$\mathbf{CP}^n$$ via the linear system $$\left|\Delta_0+\frac{n-1+d}{2}\Gamma\right|$$ where $$\Gamma$$ is a fibre of $$\mathbf{F}_d\to\mathbf{CP}^1$$ and $$\Delta_0$$ is the unique section with square $$-d$$.

In this case, if $$i\colon W\to\mathbf{CP}^n$$ is the inclusion, we have $K_X=\Phi_K^*H=(i\circ f)^*H=f^*\left(\Delta_0+\frac{n-1+d}{2}\Gamma\right)$ and $$K_W=-2\Delta_0-(d+2)\Gamma$$, so $$R=f^*\left(3\Delta_0+\frac{n+3d+3}{2}\Gamma\right)$$. Therefore the branch locus $$B$$ is homologous to $$6\Delta_0+(n+3d+3)\Gamma$$. Note that since $$B$$ has no multiple component, its intersection number with $$\Delta_0$$ must be at least $$-d$$ (it equals $$-d$$ if $$B=\Delta_0$$ and it increases for every other component of $$B$$). This means $$\Delta_0\cdot B=n-3d+3\geq -d$$ or $$d\leq \frac{1}{2}(n+3)$$.

When $$d\leq (n+3)/3$$ the generic curves in the class $$B=6\Delta_0+(n+3+3d)\Gamma$$ are smooth and irreducible. When $$(n+3)/2\geq d>(n+3)/3$$, the generic curves in the class $$B$$ are reducible of the form $$\Delta_0+B_0$$ where $$B_0$$ is an irreducible smooth curve which intersects $$\Delta_0$$ transversely at $$n+3-2d$$ points. In particular, when $$n=2d-3$$, $$\Delta_0$$ and $$B_0$$ are disjoint.

Type ($$d'$$)

($$n\geq 3$$) The surface $$W$$ is the cone $$C_{n-1}$$ on a rational curve of degree $$d=n-1$$ in $$\mathbf{CP}^{n-1}\subset\mathbf{CP}^n$$. In this case, $$f$$ factors through the minimal resolution $$\mathbf{F}_{n-1}\to C_{n-1}$$ (Lemma 1.5). The branch locus in $$\mathbf{F}_{n-1}$$ is homologous to $$6\Delta_0+4n\Gamma$$ and, again, $$\Delta_0\cdot B\geq -(n-1)$$. This means $$n\leq 5$$ (in fact the only possibilities are $$n=3,4,5$$).

Fintushel and Stern on Horikawa surfaces

Fintushel and Stern, in their classic paper on rational blowdown, study a family of Horikawa surfaces they call $$H(N)$$ with $$K^2=2N-6$$. This has $$p_g=N-1$$, so $$n=N-2$$. They take a surface of type (0), which is then a branched cover of $$\mathbf{F}_0$$ with branch locus $$B=6\Delta_0+(N+1)\Gamma$$. They point out that, smoothly, this is the same as taking the branched cover of $$\mathbf{F}_{N-3}$$ branched over $$4(\Delta+\Gamma)+2\Delta_0$$ (where $$\Delta=\Delta_0+(N-3)\Gamma$$ is the class of a positive section). At least this makes sense when $$N$$ is odd so that $$\mathbf{F}_{N-3}$$ is diffeomorphic to $$\mathbf{F}_0$$.

Note that the choice of Hirebruch surface they give would correspond to type (N-3) in Horikawa's terminology, which is actually forbidden (the class $$4(\Delta+\Gamma)+2\Delta_0$$ has no reduced holomorphic representative in $$\mathbf{F}_{N-3}$$ as its intersection with $$\Delta_0$$ is too negative). Nonetheless, this rational blow-down construction does have a complex-geometric interpretation due to Lee and Park. More on this below.

Their motivation for choosing this Hirzebruch surface is that they can find a rational homology ball $$B_{N-2,1}$$ in $$\mathbf{F}_{N-3}$$ whose complement contains the branch curve. One can see this explicitly in an almost toric picture for the Hirzebruch surface:

Figure 1: An almost toric picture of the Hirzebruch surface $$\mathbf{F}_{N-3}$$: we have performed a nodal trade at the bottom left corner so that a neighbourhood of the dashed line is a rational homology ball $$B_{N-2,1}$$ (the numbers $$(N-2,1)$$ can be read off from how the dashed line intersects the sloping edge). Note that for this picture to make sense, we need the length of $$\Gamma$$ to be strictly bigger than 1, otherwise the dashed line (with slope 1) will not intersect the sloping edge at an internal point. The length of the long bottom edge is $$N-2$$.

In the branched double cover we can therefore find two disjoint rational homology balls (note that $$N-2$$ is odd, and $$\pi_1(B_{N-2,1})$$ is cyclic of order $$N-2$$, so the rational homology ball lifts in two different ways to the double cover). Fintushel and Stern prove that the result of rationally blowing down these two rational homology balls in $$H(N)$$ is an elliptic surface $$E(N)$$.

Lee and Park on Horikawa surfaces

In an attempt to understand Fintushel and Stern's construction algebro-geometrically, Lee and Park showed that one can construct Horikawa surfaces via $$\mathbf{Q}$$-Gorenstein smoothing. The idea is that there is a singular surface $$X_N$$ with two Wahl singularities of type $$\frac{1}{(N-2)^2}(1,N-3)$$ such that

• the minimal resolution of $$X_N$$ is an elliptic surface,
• $$X_N$$ is QG-smoothable to a Horikawa surface $$H(N)$$ (in the notation of Subsection FS).
Here is how their construction works:
1. Start with a Hirzebruch surface $$\mathbf{F}_N$$. Let $$f$$ be one of the fibres, let $$\Delta_0$$ be the section with square $$-N$$ and let $$\Delta$$ be the homology class of a section with square $$N$$.
2. Find an irreducible curve $$D\subset\mathbf{F}_N$$ in the linear system $$4\Delta$$ such that:
• $$D$$ intersects $$f$$ at two points $$p$$ and $$q$$; $$D$$ has an $$A_1$$ singularity at $$p$$ and an $$A_{2N-9}$$-singularity at $$q$$. In other words, in local coordinates $$(x,y)$$ at $$p$$ (respectively $$q$$) where $$f=(x=0)$$, $$D$$ looks like $$(y-x)(y+x)=0$$ (respectively $$(y-x^{N-4})(y+x^{N-4})=0$$).
• Blow up $$\mathbf{F}_N$$ at $$p$$ and then repeatedly at $$q$$ to separate the branches of $$D$$. The fibre $$f$$ becomes a $$-2$$-curve (it had square zero and is blown up twice). The exceptional curve of the blow-up at $$q$$ is a string of $$(N-5)$$ $$(-2)$$-curves followed by a $$-1$$-curve.
Here is a picture, with $$D$$ and its proper transform drawn in blue, $$-1$$-spheres drawn dotted, and in red, a configuration of curves which, upon contraction, yields a Wahl singularity of type $$\frac{1}{(N-2)^2}(1,N-3)$$.

1. Let $$Z_N$$ be the blown-up Hirzebruch surface and let $$\tilde{D}$$ denote the proper transform of $$D$$. Let $$E$$ be the double cover of $$Z_N$$ branched over $$\tilde{D}$$. The $$-1$$-spheres become $$-2$$-spheres and there are now two disjoint red chains of spheres. The double cover $$E$$ is an elliptic surface: the preimages of the generic fibres of the Hirzebruch surface are elliptic curves (double-covering the rulings of the Hirzebruch surface branched at the four points of intersection with $$D$$). The union of all $$-2$$-spheres forms an affine $$A_{2N-6}$$ fibre (a cycle of $$-2$$-spheres).

Equivalently, we could have taken the double cover of $$\mathbf{F}_N$$ branched over $$D$$ and then $$E$$ is its minimal resolution. We can compute the invariants of $$E$$: if $$L$$ is a line bundle on $$\mathbf{F}_N$$ whose square is $$\mathcal{O}(D)$$ then $$L^2=4N$$ and $$K_{\mathbf{F}_N}\cdot L=-2N-4$$, so, using the formulae from Barth-Hulek-Peters-Van de Ven (Section 22, Eqs 9): \begin{align} K_E^2&=2K_{\mathbf{F}_N}^2+4K_{\mathbf{F}_N}\cdot L+2L\cdot L\\ &=16-8N-16+8N\\ &=0\\ c_2(E)&=2c_2(\mathbf{F}_N)+2K_{\mathbf{F}_N}\cdot L+4L\cdot L\\ &=8-4N-8+16N\\ &=12N\\ \chi(E)&=2\chi(\mathbf{F}_N)+\frac{1}{2}K_{\mathbf{F}_N}\cdot L+\frac{1}{2}L\cdot L\\ &=2-N-2+2N\\ &=N\\ p_g&=p_g(\mathbf{F}_N)+h^0(\mathbf{F}_N,K_{\mathbf{F}_N}\otimes L)\\ &=0+h^0(\mathbf{F}_N,(N-2)f)\\ &=N-1\\ q(E)&=p_q-\chi+1=0. \end{align}

2. Contracting the red chains of spheres we obtain a singular surface $$X_N$$ which Lee and Park prove admits a $$\mathbf{Q}$$-Gorenstein smoothing. Indeed, if we contract the spheres in $$Z_N$$, we get a singular surface $$Y_N$$ which is QG-smoothable; the smoothing of $$X_N$$ is obtained by taking a fibrewise branched double cover of this QG-smoothing of $$Y_N$$.

The most interesting part of this is that the singular surface obtained by contracting only one of the two red chains does not admit a $$\mathbf{Q}$$-Gorenstein smoothing. Indeed, Fintushel and Stern already pointed out that the smooth 4-manifold (in their paper called $$Y(N)$$) obtained by rationally blowing down one of the chains (which would be a model for this smoothing) admits no complex structure (Corollary 7.5):
• it is minimal (by examining its Donaldson invariants),
• it has $$c_1^2=N-3$$ and $$c_2=11N+3$$, so it is neither an elliptic surface (since $$c_1^2$$ is positive) nor a surface of general type (since the Chern numbers violate Noether's inequality), but its geometric genus is $$N-1$$ which is positive, so that exhausts the possibilities for a minimal complex surface with these invariants (by the Enriques-Kodaira classification).

Note that the surface $$X_N$$ is KSBA-stable (in other words, $$K_{X_N}$$ is ample). To see this, we appeal to the Nakai-Moishezon/Kleiman criterion: suppose that there is an irreducible curve $$C'\subset X_N$$ such that $$K_{X_N}\cdot C'\leq 0$$ and let $$C$$ be its proper transform in $$E$$. We have $K_{X_N}\cdot C'=K_E\cdot C-\sum a_jE_j\cdot C-\sum a_jE'_j\cdot C,$ where $$E_1,\ldots,E_{N-3}$$ and $$E'_1,\ldots,E'_{N-3}$$ are the Wahl strings and $$a_j$$ are the discrepancies (if $$E_1^2=-N$$, $$E_j^2=-2$$ for $$j\geq 2$$ then $$a_1=-(N-2)/(N-1)$$, $$a_2=-(N-3)/(N-1)$$,... $$a_1=-1/(N-1)$$). We also have $$K_E\cdot C\geq 0$$ with equality if and only if $$C$$ is contained in a fibre of the elliptic fibration. All the terms in $$K_{X_N}\cdot C'$$ are therefore nonnegative. Moreover, they cannot all vanish: if $$K_E\cdot C=0$$ then $$C$$ is contained in a fibre and so it either:
• intersects the sections $$E_1$$ and $$E'_1$$ positively, or
• coincides with one of the double covers of a $$-1$$-sphere, and hence intersects $$E_2$$ and $$E'_2$$ or $$E_{N-3}$$ and $$E'_{N-3}$$ positively.
In either case, there is a positive term in $$K_{X_N}\cdot C$$.

The famous open problem

The "most interesting case" from the points of view of low-dimensional topology is the case $$K^2=16\ell$$, $$p_g=8\ell+2$$ ($$n=8\ell+1$$). In this case we have two deformation classes of Horikawa surfaces, and we know that the corresponding surfaces are homotopy equivalent. In fact, all known smooth 4-manifold invariants fail to distinguish them, and it is a tantalising open problem to determine whether or not they are diffeomorphic.

• One (type (0)) is given by the branched double cover of $$\mathbf{F}_0=S^2\times S^2$$ branched along a smooth, connected curve in the homology class $$6\Delta_0+(n+3)\Gamma$$. In Fintushel-Stern notation from the previous subsection, this is $$H(8\ell+3)$$.
• The other (type (4\ell+2)) is a branched cover of $$\mathbf{F}_{4\ell+2}$$ branched over a disconnected curve $$\Delta_0\cup B_0$$ where $$B_0=5(\Delta_0+(4\ell+2)\Gamma)=5\Delta$$.
One can also take the canonical symplectic form on these and ask if they are symplectomorphic. The answer is not known; see Auroux's paper on Horikawa surfaces for more discussion about this symplectic version of the question focusing on the smallest case $$\ell=1$$, $$K^2=16$$.

Comments, corrections and contributions are very welcome; please drop me an email at j.d.evans at lancaster.ac.uk if you have something to share.

CC-BY-SA 4.0 Jonny Evans.