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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 $$\text{Jac(C)}$$ $\rho_{C,2}\, :\, \text{Gal}(K^{\text{sep}}/K) \to \text{Sp} (\text{Jac}(C)(K^{\text{sep}}) \cong \text{Sp}_6({\mathbb F}_2)$ 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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##### References:
  Beauville, A, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4), 10, 309-391, (1977) · Zbl 0368.14018  Beauville, A, Determinantal hypersurfaces, Michigan Math. J., 48, 39-64, (2000) · Zbl 1076.14534  Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3, p. 21. Springer, Berlin (1990)  Bruin, N.: Success and challenges in determining the rational points on curves. In: Proceedings of the Tenth Algorithmic Number Theory Symposium. The Open Book Series, vol. 1.1, pp. 187-212 (2013) · Zbl 1344.11048  Bruin, N., Poonen, B., Stoll, M.: Generalized explicit descent and its application to curves of genus 3. arXiv:1205.4456 · Zbl 1408.11065  Catanese, F, Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications, Invent. Math., 63, 433-465, (1981) · Zbl 0472.14024  Cook, RJ; Thomas, AD, Line bundles and homogeneous matrices, Q. J. Math. Oxf. Ser. (2), 30, 423-429, (1979) · Zbl 0437.14004  Dixon, AC, Note on the reduction of a ternary quantic to a symmetric determinant, Proc. Camb. Philos. Soc., 11, 350-351, (1902) · JFM 33.0140.04  Dixon, J.D., Mortimer, B.: Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer, New York (1996) · Zbl 0951.20001  Dye, RH, Interrelations of symplectic and orthogonal groups in characteristic two, J. Algebra, 59, 202-221, (1979) · Zbl 0409.20033  Dolgachev, I.V.: Classical Algebraic Geometry—A Modern View. Cambridge University Press, Cambridge (2012) · Zbl 1252.14001  Edge, WL, Determinantal representations of $$x^4 + y^4 + z^4$$, Math. Proc. Camb. Philos. Soc., 34, 6-21, (1938) · Zbl 0018.10003  Fried, M.D., Jarden, M.: Field arithmetic. In: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 3rd edn, vol. 11. Springer, Berlin (2008) · Zbl 1145.12001  Gross, B.H., Harris, J.: On Some Geometric Constructions Related to Theta Characteristics. Contributions to Automorphic Forms, Geometry, and Number Theory, vol. 279-311. Johns Hopkins University Press, Baltimore, MD (2004) · Zbl 1072.14032  Grothendieck, A.: Le, groupe de Brauer III. Dix Exposés sur la Cohomologie des Schémas, North-Holland, pp. 88-188. Amsterdam, Masson, Paris (1968)  Harris, J, Galois groups of enumerative problems, Duke Math. J., 46, 685-724, (1979) · Zbl 0433.14040  Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977) · Zbl 0367.14001  Hesse, O, Über elimination der variabeln aus drei algebraischen gleichungen von zweiten graden, mit zwei variabeln, J. Reine Angew. Math., 28, 68-96, (1844) · ERAM 028.0817cj  Hesse, O, Über die doppeltangenten der curven vierter ordnung, J. Reine Angew. Math., 49, 279-332, (1855) · ERAM 049.1317cj  Ho, W.: Orbit parametrizations of curves. Ph.D. Thesis, Princeton University (2009) · Zbl 0368.14018  Ishitsuka, Y.: Orbit parametrizations of theta characteristics on hypersurfaces over arbitrary fields. arXiv:1412.6978  Ishitsuka, Y., Ito, T.: The local-global principle for symmetric determinantal representations of smooth plane curves in characteristic two. arXiv:1412.8343 · Zbl 1368.14046  Ishitsuka, Y., Ito, T., On the symmetric determinantal representations of the Fermat curves of prime degree. arXiv:1412.8345. To appear in Int. J. Number Theory · Zbl 1415.11062  Jouanolou, J.-P.: Théorèmes de Bertini et Applications. Progress in Mathematics, vol. 42. Birkhäuser Boston Inc, Boston (1983) · Zbl 0519.14002  Kleiman, S.L.: The Picard Scheme, Fundamental Algebraic Geometry. Mathematical Surveys Monographs, vol. 123. American Mathematical Society, Providence (2005)  Liu, Q.: Algebraic Geometry and Arithmetic Curves. Translated from the French by Reinie Erné. Oxford Graduate Texts in Mathematics, vol. 6. Oxford Science Publications, Oxford University Press, Oxford (2002) · Zbl 0339.14020  Meyer-Brandis, T.: Berührungssysteme und symmetrische Darstellungen ebener Kurven. Diplomarbeit, Johannes Gutenberg-Universität Mainz, Mainz (1998)  Mumford, D, Theta characteristics of an algebraic curve, Ann. Sci. École Norm. Sup. (4), 4, 181-192, (1971) · Zbl 0216.05904  Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften, vol. 323, 2nd edn. Springer, Berlin (2008) · Zbl 1136.11001  Room, T.G.: The Geometry of Determinantal Loci. Cambridge University Press, Cambridge (1938) · Zbl 0020.05402  Serre, J.-P.: Abelian $$ℓ$$-adic representations and elliptic curves. McGill University Lecture Notes Written with the Collaboration of Willem Kuyk and John Labute W. A. Benjamin, Inc., New York (1968)  Serre, J.-P.: Local Fields. Graduate Texts in Mathematics, vol. 67. Springer, New York (1979)  Shioda, T, Plane quartics and Mordell-Weil lattices of type $$E_7$$, Comment. Math. Univ. St. Paul., 42, 61-79, (1993) · Zbl 0790.14025  Tate, J.T.: Global class field theory. In: Cassels, J.W.S., Fröhlich, A. (eds.) Algebraic Number Theory. Proceedings of the Instructional Conference, Brighton pp. 162-203 (1965) · Zbl 1179.11041  Tyurin, AN, On intersection of quadrics, Russ. Math. Surv., 30, 51-105, (1975) · Zbl 0339.14020  Vinnikov, V, Complete description of determinantal representations of smooth irreducible curves, Linear Algebra Appl., 125, 103-140, (1989) · Zbl 0704.14041  Vinnikov, V, Self-adjoint determinantal representations of real plane curves, Math. Ann., 296, 453-479, (1993) · Zbl 0789.14029  Wall, CTC, Nets of quadrics and theta-characteristics of singular curves, Philos. Trans. R. Soc. Lond. Ser. A, 239, 229-269, (1978) · Zbl 0382.14011
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