A complete curve of genus 105 in the moduli space of curves of genus 3

Technical Summary: A Complete Curve of Genus 105 in the Moduli Space of Curves of Genus 3

Problem Statement
For $g \geq 2$, let $\mathcal{M}_g$ denote the coarse moduli space of smooth projective curves of genus $g$ over the complex field. While $\mathcal{M}_g$ is a quasi-projective variety of dimension $3g-3$, it is never complete. A central question, posed by Harris [Har84], concerns the largest dimension of a projective subvariety $B \subset \mathcal{M}_g$. For $g=3$, it is known that this dimension is at most 1, implying the existence of complete curves. However, while the existence of such curves can be proven via intersection theory with the boundary of the Satake compactification, explicit constructions have been elusive. As noted in [Zaa05], despite the theoretical ubiquity of these curves, no one had explicitly written down a complete curve in $\mathcal{M}_3$ prior to this work. Furthermore, existing geometric constructions for $g=3$ were expected to yield base curves of "huge" genus, leaving the minimal genus of such a curve unknown.

Methodology
The author employs a strategy based on theta functions and modular forms, deviating from previous approaches that worked directly with modular forms for $\mathcal{A}_3 = \mathbb{H}_3/\Gamma_3(1)$. Instead, the construction takes place on the covering space $\mathcal{A}_3(2, 4) = \mathbb{H}_3/\Gamma_3(2, 4)$, which possesses a level structure defined by the congruence subgroup $\Gamma_3(2, 4)$.

The core of the methodology involves the following steps:

1. Embedding via Theta Constants: The space $\mathcal{A}_3(2, 4)$ is mapped into $\mathbb{P}^7$ using the 8 theta constants of the second order, $\theta_2[\sigma](\tau)$. The image of the Satake compactification is a hypersurface $X \subset \mathbb{P}^7$ defined by an irreducible equation of degree 16, derived from a specific quartic relation among the theta constants.
2. Intersection with Hyperplanes: To construct a curve, the author intersects the hypersurface $X$ with 5 randomly chosen hyperplanes $H_1, \dots, H_5$ in $\mathbb{P}^7$. The resulting variety $D = X \cap \bigcap H_i$ is a candidate for a complete curve.
3. Verification of Jacobian Locus: A critical step is ensuring that $D$ lies within the Jacobian locus (i.e., corresponds to Jacobians of genus 3 curves). According to [GDH91], a point in $\mathcal{A}_3(2, 4)$ corresponds to a non-hyperelliptic Jacobian if and only if exactly zero even theta constants vanish. The author verifies that for the chosen hyperplanes, no two distinct theta constants vanish simultaneously on $D$. This is achieved by intersecting $D$ with the quadratic relations $R_{\varepsilon, \varepsilon'} = 0$ that express squares of first-order theta constants in terms of second-order ones.
4. Quotient by the Galois Group: The curve $D$ is defined over the cover $\mathcal{A}_3(2, 4)$. To descend to the moduli space $\mathcal{M}_3$, the author computes the image $C \subset \mathcal{A}_3$ under the Galois quotient map $\psi: \mathcal{A}_3(2, 4) \to \mathcal{A}_3$, induced by the group $G = \Gamma_3(1)/\Gamma_3(2, 4)$ of order 92,897,280. The degree of the map $D \to C$ is determined by the stabilizer of the plane $P = \bigcap H_i$ under the action of $G$.
5. Descent to $\mathcal{M}_3$: Finally, the author utilizes the Torelli map extension $j: \overline{\mathcal{M}}_3 \to \overline{\mathcal{A}}_3$. Since $C$ lies in the Jacobian locus, its preimage $B = j^{-1}(C)$ is a complete curve contained in $\mathcal{M}_3$.

Key Results

• Explicit Construction: The paper provides an explicit model for a complete curve $D$ in $\mathbb{P}^7$. By choosing specific linear forms $h_1, \dots, h_5$ (e.g., $h_1 = T_1 - T_6 - T_7 + 2T_8$, etc.), the intersection $D$ is shown to be a non-singular, irreducible plane curve.
• Genus Calculation: The curve $D$ is a complete intersection of a degree 16 hypersurface and 5 hyperplanes in $\mathbb{P}^7$. Its genus is calculated as $(16-1)(16-2)/2 = 105$.
• Stabilizer Triviality: Formal computation of the action of the generators of $\Gamma_3(1)$ (inversion $S$, diagonal translations $D_k$, and crossed translations $D_{kl}$) on the vector space spanned by the theta constants reveals that the stabilizer of the defining plane $P$ is trivial (order 1). Consequently, the map $D \to C$ is an isomorphism, and $C$ is a complete curve of genus 105 in $\mathcal{A}_3$.
• Main Theorem: There exists a complete curve of genus 105 in the coarse moduli space $\mathcal{M}_3$ of smooth projective curves of genus 3 over $\mathbb{C}$.

Significance and Claims
The primary contribution of this work is the first explicit construction of a complete curve in $\mathcal{M}_3$, addressing a gap noted in the literature where such curves were known to exist theoretically but not explicitly.

• Upper Bound: The paper establishes 105 as an upper bound for the minimal genus of a complete curve in $\mathcal{M}_3$. The author explicitly states that it is unknown whether this bound is sharp, noting only that the genus must be greater than 1.
• Explicitness: Unlike previous constructions for $g \geq 4$ which often rely on curves with many automorphisms or abstract geometric arguments, this construction is explicit, providing a smooth plane curve model of degree 16.
• Modular Interpretation: The construction allows for a natural modular interpretation, enabling the construction of a genus 3 curve above each point of the base curve $D$.
• Characteristic: While the proof utilizes classical theta constants over $\mathbb{C}$, the author notes that the relations hold in positive characteristic, suggesting the result likely extends to sufficiently large characteristics.

The work relies on computational algebra (using Magma) to verify the non-vanishing of theta constant pairs and the triviality of the group stabilizer, bridging the gap between theoretical existence and explicit geometric realization.