[Paper Review] Locus of indeterminacy of the Prym map

Vitaly Vologodsky|ArXiv.org|Mar 26, 2001

Mathematical Dynamics and Fractals3 citations

TL;DR

This paper provides a simplified characterization of the locus of indeterminacy of the extended Prym map in terms of dual graphs of stable curves with involution, showing that it coincides exactly with degenerations of Friedman-Smith examples featuring four or more edges. The key result establishes that indeterminacy occurs precisely when the dual graph contains two disjoint, equivariant subgraphs connected by at least four ordinary edges, with no bold path between them.

ABSTRACT

We provide an easy characterization for the locus of indeterminacy of the Prym map in terms of the dual graphs of stable curves. As a corollary, we show that the closure of the Fridman-Smith locus coincides with the locus of indeterminacy of the Prym map.

Motivation & Objective

• To provide a simpler, combinatorial characterization of the locus of indeterminacy of the extended Prym map in terms of dual graphs of stable curves with involution.
• To clarify the relationship between the indeterminacy locus and the closure of the Friedman-Smith locus.
• To re-express the indeterminacy condition using dual graph structure, replacing complex algebraic conditions with geometric-combinatorial criteria.
• To establish equivalence between the absence of degenerations of 2-edge Friedman-Smith examples and the validity of condition (**), which ensures canonical Prym embedding.

Proposed method

• Uses dual graphs of stable curves with involution to represent the topological and combinatorial structure of degenerate curves.
• Introduces the concept of 'bold' vertices and edges (fixed by involution) and defines the subgraph $ B( ilde{\Gamma}) $ as the union of all bold vertices and edges.
• Applies a decomposition argument: if two disjoint equivariant subgraphs are connected by at least four ordinary edges and no bold path, the curve is a degeneration of a Friedman-Smith example with ≥4 edges.
• Employs topological decomposition of the graph complement to construct global equivariant subgraphs $ \Gamma'_1 $ and $ \Gamma'_2 $, preserving the required edge connectivity.
• Relies on the fact that the indeterminacy locus is closed under degeneration, so if a map is undefined on a family, it remains undefined on its limit points.
• Uses the equivalence of combinatorial conditions (*) and (**), showing that condition (**) fails precisely when the curve is a degeneration of a 2-edge Friedman-Smith example.

Experimental results

Research questions

• RQ1 When is the extended Prym map indeterminate on the moduli space $ \bar{R}_g $ of stable curves with involution?
• RQ2 How can the locus of indeterminacy be characterized using the dual graph of a stable curve with involution?
• RQ3 What is the precise relationship between the closure of the Friedman-Smith locus and the indeterminacy locus of the Prym map?
• RQ4 Under what conditions does condition (**) from [ABH] hold, and how does it relate to the absence of 2-edge degenerations?
• RQ5 Can the indeterminacy locus be fully described by a simple combinatorial condition on the dual graph, independent of complex algebraic conditions?

Key findings

• The locus of indeterminacy of the extended Prym map is exactly the closure of the Friedman-Smith locus with at least four edges.
• A curve with involution lies in the indeterminacy locus if and only if its dual graph contains two disjoint, equivariant subgraphs connected by at least four ordinary edges and no bold path between them.
• Degenerations of Friedman-Smith examples with four or more edges are precisely the curves in the indeterminacy locus, as the Prym map is undefined on the original examples and the locus is closed.
• Condition (*) holds for the indeterminacy locus, and condition (**) holds precisely when the curve is not a degeneration of a 2-edge Friedman-Smith example.
• The absence of edges of type (2) in the dual graph is equivalent to condition (**), and this implies that the Prym variety admits a canonical embedding into the Jacobian of the covering curve.
• The dual graph decomposition method allows a complete and combinatorial classification of the indeterminacy locus without relying on algebraic geometry of theta characteristics or toroidal compactifications.