[Paper Review] Invariants of a general branched cover of ${\bf P}^{1}$

Gabriel Bujokas, Anand Patel|arXiv (Cornell University)|Apr 15, 2015

Algebraic Geometry and Number Theory1 references3 citations

TL;DR

This paper investigates the syzygy bundles arising in the relative canonical embedding of a general branched cover $\alpha: C \to \mathbb{P}^1$, proving that the Casnati-Ekedahl bundle $F$ (the first syzygy bundle) is balanced for general covers when the genus $g \geq (d-3)(d-1)$. The proof uses degeneration to an admissible cover on a chain of $\mathbb{P}^1$'s and establishes a maximal rank condition for quadrics containing chains of rational normal curves.

ABSTRACT

We investigate the resolution of a general branched cover $α: C o {\bf P}^1$ in its relative canonical embedding $C \subset {\bf P} E$. We conjecture that the syzygy bundles appearing in the resolution are balanced for a general cover, provided that the genus is sufficiently large compared to the degree. We prove this for the Casnati-Ekedahl bundle, or "bundle of quadrics" $F$ - the first bundle appearing in the resolution of the ideal of the relative canonical embedding. Furthermore, we prove the conjecture for all syzygy bundles in the resolution when the genus satisfies $g = 1 \mod d$.

Motivation & Objective

• To understand the generic splitting types of syzygy bundles $N_i$ in the relative canonical embedding of a branched cover $\alpha: C \to \mathbb{P}^1$.
• To investigate whether these syzygy bundles are balanced, i.e., the difference between the largest and smallest line bundle summand degrees is at most one.
• To prove the conjecture that the bundle of quadrics $F$ (the first syzygy bundle) is balanced for general covers when the genus is sufficiently large relative to the degree.
• To establish a connection between the vanishing of $h^1(\operatorname{End} F)$ and a maximal rank problem for quadrics containing maximally connected chains of rational normal curves.

Proposed method

• Use degeneration to construct an admissible cover $\alpha: X \to P$, where $P$ is a chain of $\mathbb{P}^1$'s, to study the $F$-bundle on a singular base.
• Develop a general criterion for a vector bundle $V$ on a chain of rational curves to satisfy $h^1(\operatorname{End} V) = 0$, which implies balancedness.
• Reduce the problem of showing $h^1(\operatorname{End} F) = 0$ to a degree-2 maximal rank problem for quadrics containing a maximally connected chain of rational normal curves.
• Prove the maximal rank problem by analyzing an incidence correspondence $\Sigma$ and showing $\dim \Sigma < \dim \Sigma'$, implying the map is not dominant.
• Use dimension counting to show that generic choices of linear spaces $\Lambda_{\text{left}}, \Lambda_{\text{right}}$ do not yield the same residual intersection on the central curve $R_{\text{middle}}$.
• Leverage the fact that $F$ is the bundle of quadrics containing the fibers of the relative canonical embedding, and relate its splitting type to the geometry of the cover.

Experimental results

Research questions

• RQ1 Is the Casnati-Ekedahl bundle $F$ (the bundle of quadrics) balanced for a general branched cover $\alpha: C \to \mathbb{P}^1$ when $g \gg d$?
• RQ2 What conditions ensure that $h^1(\operatorname{End} F) = 0$, which characterizes balancedness of $F$?
• RQ3 Can the maximal rank problem for quadrics containing a maximally connected chain of rational normal curves be resolved to verify the vanishing of $h^1(\operatorname{End} F)$?
• RQ4 Does the incidence correspondence $\Sigma$ of hyperplanes and linear spaces have dimension strictly less than $\Sigma'$, implying that generic configurations do not yield the same residual intersection?
• RQ5 To what extent does the degeneration method on a chain of $\mathbb{P}^1$'s allow one to deduce generic behavior on the smooth base $\mathbb{P}^1$?

Key findings

• The Casnati-Ekedahl bundle $F$ is balanced for a general branched cover $\alpha: C \to \mathbb{P}^1$ when $g \geq (d-3)(d-1)$, confirming the first case of Conjecture A.
• The proof establishes that $h^1(\operatorname{End} F) = 0$ for such covers, which is equivalent to $F$ being balanced.
• The maximal rank problem for quadrics containing a maximally connected chain of $r$ rational normal curves is resolved by showing that the incidence correspondence $\Sigma$ has dimension $\frac{(r-1)^2}{2} + 3$, while $\Sigma'$ has dimension $\frac{(r-1)^2}{2} + 4$, so the map $\Sigma \to \Sigma'$ is not dominant.
• The construction of the admissible cover on a chain of $\mathbb{P}^1$'s allows the reduction of the smooth case to a singular degeneration, enabling the use of dimension-theoretic arguments.
• The result confirms that the bundle of quadrics $F$ is balanced in the regime $g \geq (d-3)(d-1)$, supporting the broader conjecture that syzygy bundles are generically balanced for large $g$.
• The analysis shows that for generic choices of the linear spans $\Lambda_{\text{left}}, \Lambda_{\text{right}}$, the residual intersection points on the central curve $R_{\text{middle}}$ are not the same, which rules out nontrivial solutions to the maximal rank problem.