The parity of theta characteristics is preserved by infinitesimal deformations

1. Problem Statement and Context

The paper addresses the behavior of theta characteristics (line bundles $L$ such that $L^{\otimes 2} \cong \omega_C$, where $\omega_C$ is the canonical bundle) in families of algebraic curves.

• Classical Context: It is a well-known result (proven algebraically by Mumford and generalized by Harris) that the parity of the dimension of the space of global sections, $h^0(C_t, L_t)$, is constant in a flat family of curves. Specifically, $h^0(C_t, L_t) \pmod 2$ is independent of the parameter $t$.
• The Gap: While the parity is constant on the fibers, the behavior of the restriction of the line bundle to infinitesimal neighborhoods (multiple fibers, $kC_t$) was not fully characterized in terms of parity.
• Specific Motivation: The authors were studying surfaces of general type with a canonical map of odd degree. They found that the classical invariance of parity was insufficient to rule out certain degenerate cases in their analysis. They needed to understand how the parity behaves when restricting the line bundle to non-reduced fibers (multiples of a fiber).

2. Methodology

The authors employ a combination of algebraic geometry techniques, specifically focusing on cohomology and base change, infinitesimal deformations, and linear algebra over local rings.

A. Algebraic Setup

• Base: A smooth connected curve $\Delta$ over an algebraically closed field $K$ ($\text{char}(K) \neq 2$).
• Family: A projective flat morphism $\pi: X \to \Delta$ with reduced connected fibers $C_t$.
• Bundle: A line bundle $L$ on $X$ such that $L_t^{\otimes 2} \cong \omega_{C_t}$ for all $t$.
• Local Analysis: The proofs are local, working near a point $\bar{t} \in \Delta$. They utilize the local ring $A = \mathcal{O}_{\Delta, \bar{t}}$ and its quotients $B_k = A/s^kA$ (where $s$ is a local parameter) to study the $k$-th infinitesimal neighborhood of the fiber.

B. Key Technical Tools

1. Skew-Symmetric Matrices and Rank:
   • The authors construct a non-degenerate skew-symmetric bilinear form on a vector space derived from the cohomology of the bundle.
   • They prove a lemma (Lemma 2.1) showing that for a skew-symmetric matrix $M$ over a local ring, the rank of the induced map on the quotient modules $B_k^q$ forms a non-decreasing sequence of even integers.
2. Isotropic Subspaces:
   • They utilize the intersection of two maximal isotropic subspaces ($W_1, W_2$) within a free module equipped with a symmetric bilinear form.
   • Lemma 2.2 establishes an infinitesimal version of the permanence of parity for the dimension of the intersection of these subspaces. Specifically, the sequence $kq_1 - q_k$ (where $q_k$ is the dimension of the intersection modulo $s^k$) is a non-decreasing sequence of even numbers.
3. Cohomology and Base Change:
   • Using Lemma 2.3 (based on Mumford's work), they represent the cohomology of the family as the cohomology of a two-term complex of locally free modules. This allows them to relate the cohomology of the fiber $C_t$ to the cohomology of the multiple fibers $kC_t$.
4. Relative Construction:
   • They construct a relative version of the classical diagram used by Mumford and Harris, involving the line bundle $L$ twisted by a divisor $D$ (sum of smooth points). This creates an exact diagram relating $H^0$ and $H^1$ groups over the base.

3. Key Results

Theorem 1.1 (Main Theorem)

Let $\pi: X \to \Delta$ and $L$ be as defined above. If $L_t^{\otimes 2} \cong \omega_{C_t}$ for all $t$, then for any integer $k \ge 1$ and any $t \in \Delta$:
$$h^0(kC_t, L|_{kC_t}) \equiv k \cdot h^0(C_t, L_t) \pmod 2$$
Implication: The parity of the sections on the $k$-th infinitesimal neighborhood is determined entirely by the parity on the reduced fiber.

• Refinement: The difference $k \cdot h^0(C_t, L_t) - h^0(kC_t, L|_{kC_t})$ forms a non-decreasing sequence of even numbers as $k$ increases. This is viewed as an infinitesimal version of semicontinuity.

Corollary 1.2 (Structure of Torsion)

Under the same assumptions, the torsion subsheaf of the first higher direct image sheaf $R^1\pi_*L$ splits as a direct sum of two isomorphic sheaves:
$$\text{Tor}(R^1\pi_*L) \cong \mathcal{T} \oplus \mathcal{T}$$
Derivation: The proof analyzes the rank of the torsion sheaf at a point. By combining the constancy of parity in families with Theorem 1.1, they show that the multiplicities of the torsion components must be even, forcing the splitting structure.

4. Significance and Applications

1. Independent Interest: The result provides a deeper understanding of the deformation theory of theta characteristics, extending the classical invariance theorem to the infinitesimal level.
2. Application to Surface Theory: The primary motivation was the classification of surfaces of general type with a canonical map of odd degree.
   • In the analysis of such surfaces, certain degenerate cases (where the canonical map behaves irregularly) could not be ruled out using only the classical parity theorem.
   • Corollary 1.2 provided the necessary constraint (the even splitting of the torsion) to eliminate these specific pathological cases, allowing the authors to complete their classification (referenced as [MLPP26] in the text).
3. Methodological Contribution: The paper demonstrates a robust technique for translating geometric parity questions into linear algebra problems over local Artinian rings (using skew-symmetric matrices), which can be applied to other problems involving infinitesimal deformations of moduli spaces.

Summary

The paper proves that the parity of the dimension of global sections of a theta characteristic is preserved not just on the fibers of a family, but also on their infinitesimal neighborhoods. This leads to a structural result regarding the torsion of the higher direct image sheaf, which is crucial for the classification of specific types of algebraic surfaces. The proof elegantly bridges the gap between the geometry of line bundles and the linear algebra of skew-symmetric forms over local rings.