Log canonical models and variation of GIT for genus four canonical curves

To put our results in context, we recall some background on the Hassett–Keel program. Namely, for $\alpha\in[0,1]$, the log minimal models of $\overline{M}_{g}$ are defined to be the projective varieties

where $\delta$ is the boundary divisor in $\overline{M}_{g}$. Hassett and Hyeon have explicitly constructed the log minimal models $\overline{M}_{g}(\alpha)$ for $\alpha\geq\frac{7}{10}-\epsilon$ (see [HH09, HH08]). Hyeon and Lee have also described the next stage of the program in the specific case that $g=4$ (see [HL10a]): as $\alpha$ decreases from $\frac{2}{3}+\epsilon$ to $\frac{2}{3}$, they construct a map that contracts the locus of Weierstrass genus 2 tails, replacing them with $A_{4}$ singularities. Thus, the known spaces for the Hassett–Keel program in genus $4$ are:

where the notation $\overline{M}_{g}(I)$ for an interval $I$ means $\overline{M}_{g}(\alpha)\cong\overline{M}_{g}(\beta)$ for all $\alpha,\beta\in I$. The double arrows correspond to divisorial contractions, the single arrows to small contractions, and the dashed arrows to flips.

The main result of the paper is the construction of the log minimal models $\overline{M}_{4}(\alpha)$ for $\alpha\leq\frac{5}{9}$ via a VGIT analysis of canonically embedded curves in $\mathbb{P}^{3}$.

Thus in genus $4$, the remaining unknown range for the Hassett–Keel program is the interval $\alpha\in(\frac{5}{9},\frac{2}{3})$. Using the geometric meaning of the spaces $\overline{M}_{4}(\alpha)$ for $\alpha\leq\frac{5}{9}$ and the predictions of [AFS10], we expect that there are exactly two more critical values: $\alpha=\frac{19}{29}$, when the divisor $\delta_{2}$ should be contracted to a point, and $\alpha=\frac{49}{83}$, when the locus of curves with hyperelliptic normalization obtained by introducing a cusp at a Weierstrass point should be flipped, being replaced by curves with $A_{6}$ singularities. We do not expect that these models can be obtained by further varying the GIT problem we consider here. In fact, since each of these predicted models arises prior to the predicted flip of the hyperelliptic locus ($\alpha=\frac{5}{9}$), they should be unrelated to spaces of canonical curves. It is believed that each of these two intermediate models ought to correspond to a quotient of the Hilbert scheme of bicanonical curves.

As already mentioned, GIT for pluricanonical curves has long been used to produce projective models for the moduli space of curves. For example Mumford used asymptotic stability for $\nu$-canonical curves, with $\nu\geq 5$, to show the projectivity of $\overline{M}_{g}$, and recently the case $\nu<5$ has been used in the Hassett–Keel program. The basic idea is that as the values $\nu$ and $m$ decrease one should obtain the log minimal models $\overline{M}_{g}(\alpha)$ for progressively smaller values of $\alpha$ (e.g. [FS10, Table 1]). Thus from the perspective of the Hassett–Keel program, it is of interest to understand GIT for canonically embedded curves. This turns out to be difficult, and to our knowledge the only case where the stability conditions have been described completely prior to this paper is for genus $3$. On the other hand, it was recently proved (see [AFS12]) that the generic non-singular canonical curve of arbitrary genus is stable. In this paper, we completely describe the stability conditions for genus $4$ canonical curves.

We set up the analysis of the GIT stability for canonical genus $4$ curves as follows. The canonical model of a smooth, non-hyperelliptic genus $4$ curve is a $(2,3)$-complete intersection in $\mathbb{P}^{3}$. A natural parameter space for complete intersections is a projective bundle $\mathbb{P}E\to\mathbb{P}^{9}$ on which $G=\mathrm{SL}(4,\mathbb{C})$ acts naturally. Since $\mathrm{rank}(\operatorname{Pic}(\mathbb{P}E))=2$, the GIT computation involves a choice of linearization parameterized by $t\in\mathbb{Q}_{+}\cup\{0\}$ (corresponding to the linearization $\eta+th$, where $\eta$ is the pullback of $\mathscr{O}(1)$ from the space of quadrics $\mathbb{P}^{9}$ and $h$ is the relative $\mathscr{O}(1)$). In this paper, we analyze the geometry of the quotients $\mathbb{P}E/\!\!/_{t}\mathrm{SL}(4)$ as the linearization varies and relate them to the Hassett–Keel spaces $\overline{M}(\alpha)$. We note that a related setup for GIT for complete intersections occurs in recent work of Benoist [Ben11].

One naturally identifies two special cases. First, for $0<t\ll 1$ one easily sees that $\mathbb{P}E/\!\!/_{t}\mathrm{SL}(4)$ coincides with the GIT quotient for $(3,3)$ curves on $\mathbb{P}^{1}\times\mathbb{P}^{1}$; this was analyzed by Fedorchuk [Fed12]. At the other extreme, the case $t=\frac{2}{3}$ was shown in [CMJL12] to be isomorphic to both the quotient of the Chow variety for genus $4$ canonical curves, as well as to the Hassett–Keel space $\overline{M}_{4}(\frac{5}{9})$. The content of this paper is to describe the GIT quotient for the intermediary values $t\in(0,\frac{2}{3})$. We work with $\mathbb{P}E$, but show that all of the quotients of type $\operatorname{Hilb}^{m}_{4,1}/\!\!/\mathrm{SL}(4)$ arise in this way. The advantage of working with $\mathbb{P}E$ is that we have VGIT on a single, elementary space, where the stability computation is straightforward and corresponds directly to the variation of parameters.

As mentioned, the Hassett–Keel program aims to give modular interpretations to the spaces $\overline{M}_{g}(\alpha)$ and to the birational maps between them. Essentially, as $\alpha$ decreases, it is expected that $\overline{M}_{g}(\alpha)$ parameterizes curves with increasingly complicated singularities, and at the same time special curves from $\overline{M}_{g}$ are removed (e.g. curves with elliptic tails, or hyperelliptic curves, etc.). In the situation of our main result, the maps of the diagram (0.2) are intuitively described as follows. In $\overline{M}_{4}(\frac{5}{9})$, the hyperelliptic locus is contracted, as is the locus of elliptic triboroughs, and the locus of curves in $\Delta_{0}$ with hyperelliptic normalization obtained by gluing two points that are conjugate under the hyperelliptic involution. The next map flips these loci, replacing them with curves that have $A_{8}$, $D_{4}$, and $A_{7}$ singularities, respectively.

The second flip (at $\alpha=\frac{23}{44}$) removes the locus of cuspidal curves whose normalization is hyperelliptic, replacing them with curves possessing a separating $A_{7}$ singularity. The third flip (at $\alpha=\frac{1}{2}$) removes the locus of nodal curves whose normalization is hyperelliptic, replacing them with the union of a conic and a double conic. Finally, the map to $\overline{M}_{4}(\frac{29}{60})$ contracts the Gieseker–Petri divisor to a point, corresponding to a triple conic. This geometric description of the various maps is summarized in Tables 2 and 3 in §3.

We note that the critical slopes occurring in our analysis are in concordance with the general predictions of Alper–Fedorchuk–Smyth [AFS10]. We also note that at $\alpha=\frac{5}{9}$ and $\alpha=\frac{23}{44}$, we observe a phenomenon that first occurs in genus $4$. Namely, the critical values at which the separating $A_{5}$ and $A_{7}$ singularities appear differ from those at which the non-separating singularities appear.

The authors are grateful to O. Benoist and M. Fedorchuk for discussions relevant to this paper, and for specific comments on an earlier draft. We also thank the referees for detailed comments that have improved the paper.

The following table, relating the parameters $\alpha$, $t$ and $m$, describes the relationships among the various spaces occurring in this paper. Note that the following relations (see Proposition 1.8 and Theorem 7.1) hold:

We start by recalling the definition of the bundle $\mathbb{P}E$ from [CMJL12] and establishing some basic properties. We fix the notation

for each $d\in\mathbb{Z}$, and define $\mathcal{Q}$ to be the universal quadric:

There is an exact sequence of sheaves

Setting $\pi_{1}:\mathbb{P}^{3}\times\mathbb{P}V_{2}\to\mathbb{P}^{3}$ (resp. $\pi_{2}:\mathbb{P}^{3}\times\mathbb{P}V_{2}\to\mathbb{P}V_{2}$) to be the first (resp. second) projection, then tensoring (1.1) by $\pi_{1}^{*}\mathscr{O}_{\mathbb{P}^{3}}(3)$ and projecting with $\pi_{2*}$ we obtain an exact sequence:

We will define the projective bundle $\mathbb{P}E$ using the locally free sheaf on the right.

We point out that

so (1.2) can be written as

As mentioned above, there is a family

of $(2,3)$-subschemes of $\mathbb{P}^{3}$ parameterized by $\mathbb{P}E$ that is flat exactly over the locus $U$ of points $([q],[f])$ such that $q$ and $f$ do not have a common linear factor. Consequently, there is a birational map

whose restriction to $U$ is a morphism; here $\operatorname{Hilb}_{4,1}$ is the component of the Hilbert scheme containing genus $4$ canonical curves.

We now consider the nef cone and pseudoeffective cone of $\mathbb{P}E$. Benoist [Ben11] has determined the nef cones of more general spaces of complete intersections. We state a special case of his result here, together with a basic observation on the pseudoeffective cone.

Rojas–Vainsencher [RV02] have constructed an explicit resolution $W$ of the rational map $\mathbb{P}E\dashrightarrow\operatorname{Hilb}_{4,1}$, giving a diagram:

It is shown in [RV02, Thm. 3.1] that $W$ can be obtained from $\mathbb{P}E$ via a sequence of seven blow-ups along $\mathrm{SL}(4)$-invariant smooth subvarieties, and the resulting space $W$ is isomorphic to $\mathbb{P}E$ along $U\subseteq\mathbb{P}E$, the locus of complete intersections. In particular, $\mathrm{SL}(4)$ acts on $W$ (compatibly with the action on $\mathbb{P}E$), and $W$ is non-singular.

The only planar singularities of multiplicity two are the $A_{k}$ singularities. We will see later in our GIT analysis that when $k$ is odd, it is important to distinguish between two types of $A_{k}$ singularities, those that separate the curve and those that do not.

Let $C$ be a $(2,3)$-complete intersection with a singularity of multiplicity 3, which does not contain a line component meeting the residual curve only at the singularity. Notice that projection from the singularity maps $C$ onto a cubic in $\mathbb{P}^{2}$. It follows that $C$ is contained in the cone over this cubic. We choose specific coordinates so that the singular point is $p=(1,0,0,0)$ and the tangent space to the quadric at $p$ is given by $x_{3}=0$. Now, consider the 1-PS with weights $(1,0,0,-1)$. The flat limit of $C$ under this one-parameter subgroup is cut out by the equations:

where $q^{\prime}$ and $f^{\prime}$ are forms in the variables $x_{1},x_{2}$. We see that this limit is the union of three (not necessarily distinct) conics meeting at the points $p$ and $(0,0,0,1)$.

Following [Fed12] we will refer to these unions of conics as tangent cones. In our GIT analysis we will see that, for a given linearization, the semistable tangent cones are precisely the polystable (i.e. semi-stable with closed orbit) curves with triple point singularities. Note that the conics are distinct if and only if the original triple point is of type $D_{4}$.

As we vary the GIT parameters, we will see that certain subloci of curves on singular quadrics are progressively destabilized. In this section we briefly describe each of these loci. The first locus to be destabilized is the set of curves lying on low-rank quadrics.

Following [AFS10], we refer to such curves as elliptic triboroughs. The locus of elliptic triboroughs is expected to be flipped in the Hassett–Keel program at the critical value $\alpha=\frac{5}{9}$. This is exactly what we will prove in the following sections.

We now consider curves on a quadric of rank 3. More specifically, we will see that a curve lies on a quadric cone if and only if its normalization admits a Gieseker–Petri special linear series. The proposition below follows by a standard argument. The result is not needed in the ensuing proofs, but is useful in giving a geometric interpretation to the stability computations in later sections.

Let $\operatorname{Chow}_{4,1}$ denote the irreducible component of the Chow variety containing genus 4 canonical curves. In [CMJL12], the authors study the GIT quotient $\operatorname{Chow}_{4,1}/\!\!/_{\Lambda_{\infty}}\mathrm{SL}(4)$ and obtain the following:

Additionally, it is shown in [CMJL12] that the quotient of the Chow variety coincides with one of the Hassett–Keel spaces, specifically:

For the reader’s convenience, we briefly describe the birational contraction $\overline{M}_{4}\dashrightarrow\operatorname{Chow}_{4,1}/\!\!/\mathrm{SL}(4)$ in Table 2. In order to make sense of the table, we need to recall some standard terminology. Specifically, a tail of genus $i$ is a genus $i$ connected component of a curve that meets the residual curve in one point. Similarly, a bridge of genus $i$ is a genus $i$ connected component of a curve that meets the residual curve in two points. By conjugate points on a hyperelliptic curve, we mean points that are conjugate under the hyperelliptic involution. An elliptic triborough is a genus $1$ connected component of a curve that meets the residual curve in three points.

Recall that every canonically embedded curve $C$ of genus 4 is contained in a quadric in $\mathbb{P}^{3}$. If this quadric is smooth, then it is isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$, and $C$ is a member of the class $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(3,3)|$. The automorphism group of the quadric is $\mathrm{SO}(4)$, which is isogenous to $\mathrm{SL}(2)\times\mathrm{SL}(2)$. The GIT quotient $|\mathcal{O}(3,3)|/\!\!/\mathrm{SO}(4)$ was studied in detail by Fedorchuk in [Fed12]. Because this GIT quotient appears as the last stage of the log minimal model program for $\overline{M}_{4}$, we refer to curves that are (semi)stable with respect to this action as terminally (semi)stable. We summarize the results of [Fed12] here.

As mentioned above, Fedorchuk [Fed12] showed that this GIT quotient is the final non-trivial step in the Hassett–Keel program for genus $4$, specifically:

In this paper we are interested in describing the behavior of the Hassett–Keel program for genus $4$ curves in the interval $\alpha\in\left[\frac{8}{17},\frac{5}{9}\right]$ (with endpoints described by (3.8) and (3.4) respectively). In particular, in the following sections, we will give an explicit factorization of the birational map

as the composition of two flips and a divisorial contraction.

For the moment, by comparing the stability conditions given by Theorems 3.1 and 3.6 and by simple geometric considerations, we obtain a rough description of the birational map $\Psi$ as summarized in Table 3 (see also [Fed12, Table 1]). The first three lines of the table correspond to strictly semi-stable points of $\operatorname{Chow}_{4,1}$ that are all flipped by the map $\overline{M}_{4}(\frac{5}{9}-\epsilon)\to\overline{M}_{4}(\frac{5}{9})$. Then, note that every Chow-stable curve contained in a quadric cone is terminally unstable. There are three types of such curves: those that do not meet the vertex of the cone, those that meet it in a node, and those that meet it in a cusp. These correspond to the latter three lines in the table, as well as the three critical slopes in our VGIT problem. These last three lines correspond, in order, to the flip at $\alpha=\frac{23}{44}$, the flip at $\alpha=\frac{1}{2}$, and the divisorial contraction at $\alpha=\frac{29}{60}$.

Let us start by recalling the Hilbert–Mumford index for hypersurfaces. That is, we consider the case of $\mathrm{SL}(r+1)$ acting on $\mathbb{P}H^{0}(\mathbb{P}^{r},\mathscr{O}_{\mathbb{P}^{r}}(d))$. In this case, given a one-parameter subgroup (1-PS) $\lambda:\mathbb{G}_{m}\to\mathrm{SL}(r+1)$, the action on $H^{0}(\mathbb{P}^{r},\mathscr{O}_{\mathbb{P}^{r}}(1))$ can be diagonalized. We describe the action of $\lambda$ in these coordinates with a weight vector $\alpha=(\alpha_{0},\alpha_{1},\ldots,\alpha_{r})$. For a monomial $x^{a}=x_{0}^{a_{0}}\cdots x_{r}^{a_{r}}\in H^{0}(\mathbb{P}^{r},\mathscr{O}_{\mathbb{P}^{r}}(d))$ in these coordinates, we define the $\lambda$-weight of $x^{a}$ to be