[Paper Review] A small resolution for triple covers in algebraic geometry

Daniele Faenzi, Janis Stipins|ArXiv.org|Apr 18, 2002

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper constructs a small resolution for triple covers in algebraic geometry by introducing a new variety $\mathfrak{S}_X$ inside $\mathbb{P}(E^*)$, birational to the original cover $X$, which resolves singularities at fat-point ramification loci via $\mathbb{P}^1$ fibers. The resolution is defined via a global section $\sigma$ of $S^3(E)^* \otimes \Lambda^2(E)$, and $X$ embeds into a $\mathbb{P}^1$-bundle if and only if there is no fat-point ramification.

ABSTRACT

Given a triple cover p: X --> Y of varieties, we produce a new variety Z and a birational morphism f: Z --> X which is an isomorphism away from the fat-point ramification locus of p. The variety Z has a natural interpretation in terms of the data describing the triple cover, and the morphism f has an elegant geometric description.

Motivation & Objective

• To provide a geometric construction of a small resolution for triple covers $\pi: X \to Y$ that resolves singularities at fat-point ramification loci.
• To show that such a resolution exists globally using a rank 2 locally free sheaf $E$ and a global section $\sigma$ of $S^3(E)^* \otimes \Lambda^2(E)$.
• To characterize when $X$ can be embedded as a subvariety of a $\mathbb{P}^1$-bundle over $Y$ via the absence of fat-point ramification.
• To offer a simplified geometric interpretation of a more general resolution result, providing insight into small resolutions in the triple cover setting.

Proposed method

• Define $\mathfrak{S}_X$ as a subvariety of $\mathbb{P}(E^*)$ using the global section $\sigma$ of $S^3(E)^* \otimes \Lambda^2(E)$, which arises from the triple cover data.
• Construct a birational morphism $\rho_X: \mathfrak{S}_X \to X$ that is an isomorphism away from the fat-point ramification locus.
• Show that the fiber of $\rho_X$ over each fat-point ramification point is isomorphic to $\mathbb{P}^1$, making $\rho_X$ a small resolution.
• Globalize the local resolution by using Miranda’s result that triple covers are classified by $E$ and $\sigma$, ensuring compatibility across affine patches.
• Use the isomorphism between $\mathfrak{S}_X$ and the blow-up of a Weil divisor in $X$ to interpret $\rho_X$ as a blow-up along a non-Cartier divisor.
• Demonstrate that the rational map $\psi$ associated with the resolution has indeterminacy precisely at the fat-point ramification locus, confirming the resolution's support.

Experimental results

Research questions

• RQ1 Under what conditions can a triple cover $\pi: X \to Y$ be realized as a subvariety of a $\mathbb{P}^1$-bundle over $Y$?
• RQ2 How can a small resolution be constructed geometrically for triple covers, particularly at fat-point ramification loci?
• RQ3 What is the role of the global section $\sigma$ of $S^3(E)^* \otimes \Lambda^2(E)$ in defining the resolution variety $\mathfrak{S}_X$?
• RQ4 Why is the exceptional set of the resolution $\rho_X: \mathfrak{S}_X \to X$ of codimension greater than one, despite being a small resolution?
• RQ5 How does the resolution $\rho_X$ relate to the blow-up of a Weil divisor in $X$, and why is this divisor not Cartier?

Key findings

• The variety $\mathfrak{S}_X$ is a subvariety of $\mathbb{P}(E^*)$ defined by the cubic form associated to the global section $\sigma$ of $S^3(E)^* \otimes \Lambda^2(E)$.
• The morphism $\rho_X: \mathfrak{S}_X \to X$ is a small resolution: it is an isomorphism outside the fat-point ramification locus and has $\mathbb{P}^1$ fibers over such points.
• The resolution $\rho_X$ is the blow-up of a Weil divisor in $X$, which is not Cartier, hence the blow-up is not an isomorphism.
• The exceptional set of $\rho_X$ has codimension greater than one in $\mathfrak{S}_X$, confirming it is a small resolution in the algebraic geometry sense.
• A triple cover $\pi: X \to Y$ embeds into a $\mathbb{P}^1$-bundle over $Y$ if and only if it has no fat-point ramification, as such points would otherwise force a two-dimensional Zariski tangent space in the fiber.
• The rational map $\psi$ used in the construction is indeterminate precisely at the fat-point ramification locus, confirming the resolution's support is exactly there.