An explicit class of Lagrangian surfaces

TL;DR

This work constructs an explicit family of general-type Lagrangian surfaces with invariants $q=4$, $p_g=6$, $K^2=24$ by taking the quotient $X=W/K$ of the Galois closure $W$ of a degree-$3$ map from a very general $(1,6)$-polarized abelian surface $A$. The authors show $X$ is Lagrangian in its Albanese variety, and that its canonical map is a $2:1$ cover onto a degree-$12$ surface in $\mathbb P^5$ with $44$ nodes, while $X$ has an Albanese map $\alpha:X\to (A\times A)/K$ that is a local immersion with exactly two points identified. They compute the invariants of the Galois closure $W$ and its quotients, study the ramification locus (including a genus-$6$ hyperelliptic curve), and establish that $X$ admits no fibration to curves of positive genus, culminating in a detailed description of the canonical image of $X$ as a degree-$2$ cover over a $12$-nodal surface. The results connect to Schoen-type surfaces and contribute to understanding Hodge-theoretic aspects and moduli questions for such Lagrangian constructions in abelian four-folds.

Abstract

We construct a family of general type surfaces with $q=4$, $p_g=6$ and $K^2=24$. These surfaces enjoy some interesting properties: they are Lagrangian in their Albanese variety and their canonical map is $2:1$ onto a degree $12$ surface in $\mathbb P^5$ with $44$ even nodes.

Figures (3)

• Figure 1: $\tilde{\varphi}$ maps $E$ and $H_E$ to the line $l$ in the fixed point space of $-1 \colon \mathbb{P}^2 \to \mathbb{P}^2$.
• Figure 2: The quotient by $K = K(L_B)$ on the exceptional locus, the hyperelliptic curves and the ramification of $\tilde{\varphi}$.
• Figure 3: The map between the ramification of $\tilde{\varphi}$ and the branch locus of $\delta$

Theorems & Definitions (50)

• Theorem A
• Theorem B
• Theorem 2.1
• Theorem 2.2
• proof
• Remark 1
• Proposition 1
• proof
• Lemma 1
• proof