[Paper Review] Log canonical models and variation of GIT for genus four canonical curves

Sebastian Casalaina‐Martin, David Jensen+1 more|arXiv (Cornell University)|Mar 22, 2012

Algebraic Geometry and Number Theory23 references7 citations

TL;DR

This paper completes the final steps of the Hassett–Keel program for genus four curves by constructing log canonical models $¯{M}_4(\alpha)$ for $\alpha \leq \frac{5}{9}$ via a variation of GIT (VGIT) on a single parameter space $\mathbb{P}E$, which parametrizes complete intersections of a quadric and a cubic in $\mathbb{P}^3$. The authors resolve stability conditions for linearizations outside the ample cone and show that the resulting GIT quotients realize all intermediate log canonical models through a continuous family of birational transformations.

ABSTRACT

We discuss GIT for canonically embedded genus four curves and the connection to the Hassett-Keel program. A canonical genus four curve is a complete intersection of a quadric and a cubic, and, in contrast to the genus three case, there is a family of GIT quotients that depend on a choice of linearization. We discuss the corresponding VGIT problem and show that the resulting spaces give the final steps in the Hassett-Keel program for genus four curves.

Motivation & Objective

• To complete the Hassett–Keel program for genus four curves by constructing log canonical models $\overline{M}_4(\alpha)$ for $\alpha \leq \frac{5}{9}$.
• To analyze GIT stability for canonically embedded genus four curves, which are complete intersections of a quadric and a cubic in $\mathbb{P}^3$.
• To unify all GIT quotients for these curves into a single variation of GIT (VGIT) problem on the space $\mathbb{P}E$.
• To resolve the technical challenge of analyzing stability for linearizations outside the ample cone in a VGIT framework.
• To establish a precise birational correspondence between the GIT quotients $\mathbb{P}E/\!/_{t}\mathrm{SL}(4)$ and the log canonical models $\overline{M}_4(\alpha)$.

Proposed method

• Use the space $\mathbb{P}E$, a smooth, elementary, birational model of the Hilbert scheme of complete intersections of type $(2,3)$ in $\mathbb{P}^3$, as the parameter space for the VGIT problem.
• Perform a VGIT analysis on $\mathbb{P}E$ with respect to the action of $\mathrm{SL}(4)$, varying the linearization $t$ to obtain different GIT quotients.
• Handle linearizations outside the ample cone by circumventing ambiguity in Mumford’s numerical criterion through careful geometric and cohomological analysis.
• Relate the GIT quotients to the log canonical models via pullback of line bundles: $\varphi^*(4s\eta + 4h) = (34s - 33)\lambda - (4s - 4)\delta_0 - (14s - 15)\delta_1 - (18s - 21)\delta_2$.
• Use the fact that $\delta_1$ and $\delta_2$ are $\varphi$-exceptional divisors to simplify cohomology rings and establish isomorphism with $\overline{M}_4(\alpha)$.
• Leverage known results from [CMJL12] and [Fed12] for extremal cases to extend the construction to intermediate values of $\alpha$.

Experimental results

Research questions

• RQ1 How can the log canonical models $\overline{M}_4(\alpha)$ for $\alpha \leq \frac{5}{9}$ be realized as GIT quotients of canonically embedded genus four curves?
• RQ2 Can a single VGIT problem on a space $\mathbb{P}E$ unify the entire family of GIT quotients for $(2,3)$-complete intersections in $\mathbb{P}^3$?
• RQ3 What is the precise relationship between the linearization parameter $t$ and the log canonical model parameter $\alpha$ in the Hassett–Keel program?
• RQ4 How can stability be analyzed for linearizations lying outside the ample cone in a VGIT setting?
• RQ5 What is the birational geometry of the resulting GIT quotients, and how do they relate to divisorial and small contractions in the Hassett–Keel program?

Key findings

• The log canonical models $\overline{M}_4(\alpha)$ for $\alpha \leq \frac{5}{9}$ are isomorphic to the GIT quotients $\mathbb{P}E/\!/_{t}\mathrm{SL}(4)$, with $t = \frac{34\alpha - 16}{33\alpha - 14}$.
• The entire family of log canonical models for genus four curves is realized via a single VGIT problem on $\mathbb{P}E$, unifying previously separate constructions.
• The paper resolves the technical issue of analyzing stability for linearizations outside the ample cone by proving that the numerical criterion remains well-defined and effective in this context.
• The map $\varphi: \overline{M}_4 \dashrightarrow \mathbb{P}E/\!/_{t}\mathrm{SL}(4)$ is a birational contraction for all $t \in (0, \frac{2}{3}]$, and the pullback of the line bundle $4s\eta + 4h$ matches the canonical divisor $K_{\overline{M}_4} + \alpha\delta$.
• The coefficients of the pullback divisor are explicitly computed: $a = 34s - 33$, $b_0 = 4s - 4$, $b_1 = 14s - 15$, $b_2 = 18s - 21$, with $s = \frac{1}{t}$.
• The construction completes the Hassett–Keel program for genus four curves outside a small range, providing the final steps in the log minimal model program for $\overline{M}_4$.