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The local-global principle for symmetric determinantal representations of smooth plane curves. (English) Zbl 1390.14086
Let $$C\subset {\mathbb P}_K^2$$ be a smooth plane curve of degree $$n\geq 1$$ over a field $$K$$. We say that $$C$$ admits a symmetric determinantal representation over $$K$$ if there exists a triple of symmetric matrices $$(M_0,M_1,M_1)$$ of size $$n$$ with entries in $$K$$ such that $$C$$ is defined by the equation $\det (X_0,N_0+X_1M_1+X_2M_2) = 0\, .$
The paper studies the local-global principle for the existence of symmetric determinantal representations of smooth plane curves defined over a global field $$K$$ with $$\text{char}\, K\neq 2$$.
The main results of the paper are the following: \smallskip
{Theorem 1.} Let $$K$$ be a global field of characteristic different from two, and let $$C\subset {\mathbb P}_K^2$$ be a smooth plane curve of degree 2 or 3. If $$C$$ admits a symmetric determinantal representation over the completion $$K_v$$ for each place $$v$$ of $$K$$, then $$C$$ admits a symmetric determinantal representation over $$A$$. \smallskip
{Theorem 2.} Let $$K$$ be a global field of characteristic different from two, and let $$C\subset {\mathbb P}_K^2$$ be a smooth plane curve of degree 4. Assume that the associated mod 2 Galois representation on the 2-torsion points on the Jacobian variety $$\text{Jac(C)}$$ $\rho_{C,2}\, :\, \text{Gal}(K^{\text{sep}}/K) \to \text{Sp} (\text{Jac}(C)[2](K^{\text{sep}}) \cong \text{Sp}_6({\mathbb F}_2)$ is surjective. The there is a finite extension $$L/K$$ such that $$C$$ admits a symmetric determinantal representation over $$L_w$$ for each place $$w$$ of $$L$$ but not over $$L$$. \smallskip
The proof of Theorem 1 relies on the existence of a $$K$$-rational point on the conic for $$n=2$$, and of a non-trivial $$K$$-rational 2-torsion point on the Jacobian variety of the curve for $$m=3$$.
The proof of Theorem 2 makes use of a group-theoetic lemma on the action of the subgroups of $$\text{Sp}({\mathbb F}_2)$$ on quadratic forms over $${\mathbb F}_2$$.

##### MSC:
 14H50 Plane and space curves 11D41 Higher degree equations; Fermat’s equation 14K30 Picard schemes, higher Jacobians 14K15 Arithmetic ground fields for abelian varieties 14F22 Brauer groups of schemes
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