The separating semigroup of a real curve. (English. French summary) Zbl 1440.14271
Summary: We introduce the separating semigroup of a real algebraic curve of dividing type. The elements of this semigroup record the possible degrees of the covering maps obtained by restricting separating morphisms to the real part of the curve. We also introduce the hyperbolic semigroup which consists of elements of the separating semigroup arising from morphisms which are compositions of a linear projection with an embedding of the curve to some projective space.
We completely determine both semigroups in the case of maximal curves. We also prove that any embedding of a real curve of dividing type to projective space of sufficiently high degree is hyperbolic. Using these semigroups we show that the hyperbolicity locus of an embedded curve is in general not connected.
We completely determine both semigroups in the case of maximal curves. We also prove that any embedding of a real curve of dividing type to projective space of sufficiently high degree is hyperbolic. Using these semigroups we show that the hyperbolicity locus of an embedded curve is in general not connected.
MSC:
| 14P25 | Topology of real algebraic varieties |
| 14H50 | Plane and space curves |
| 14H51 | Special divisors on curves (gonality, Brill-Noether theory) |
| 20M99 | Semigroups |
References:
| [1] | Ahlfors, Lars V., Open Riemann surfaces and extremal problems on compact subregions, Comment. Math. Helv., 24, 100-134 (1950) · Zbl 0041.41102 · doi:10.1007/BF02567028 |
| [2] | Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A.; Harris, Joe, Geometry of algebraic curves. Vol. I, 267, xvi+386 p. pp. (1985), Springer · Zbl 1235.14002 |
| [3] | Basili, Bénédicte, Indice de Clifford des intersections complètes de l’espace, Bull. Soc. Math. Fr., 124, 1, 61-95 (1996) · Zbl 0871.14027 · doi:10.24033/bsmf.2276 |
| [4] | Brändén, Petter, Polynomials with the half-plane property and matroid theory, Adv. Math., 216, 1, 302-320 (2007) · Zbl 1128.05014 · doi:10.1016/j.aim.2007.05.011 |
| [5] | Choe, Young-Bin; Oxley, James G.; Sokal, Alan D.; Wagner, David G., Homogeneous multivariate polynomials with the half-plane property, Adv. Appl. Math., 32, 1-2, 88-187 (2004) · Zbl 1054.05024 · doi:10.1016/S0196-8858(03)00078-2 |
| [6] | Ciliberto, Ciro; Lazarsfeld, Robert, Complete intersections (Acireale, 1983), 1092, On the uniqueness of certain linear series on some classes of curves, 198-213 (1984), Springer · Zbl 0548.14016 · doi:10.1007/BFb0099363 |
| [7] | Coppens, Marc, The separating gonality of a separating real curve, Monatsh. Math., 170, 1, 1-10 (2013) · Zbl 1276.14039 · doi:10.1007/s00605-012-0413-x |
| [8] | Coppens, Marc; Huisman, Johannes, Pencils on real curves, Math. Nachr., 286, 8-9, 799-816 (2013) · Zbl 1307.14047 · doi:10.1002/mana.201100196 |
| [9] | Fisk, Steve, Polynomials, roots, and interlacing |
| [10] | Gabard, Alexandre, Sur la représentation conforme des surfaces de Riemann à bord et une caractérisation des courbes séparantes, Comment. Math. Helv., 81, 4, 945-964 (2006) · Zbl 1112.14066 · doi:10.4171/CMH/82 |
| [11] | Gårding, Lars, Linear hyperbolic partial differential equations with constant coefficients, Acta Math., 85, 1-62 (1951) · Zbl 0045.20202 · doi:10.1007/BF02395740 |
| [12] | Gross, Benedict H.; Harris, Joe, Real algebraic curves, Ann. Sci. Éc. Norm. Supér., 14, 2, 157-182 (1981) · Zbl 0533.14011 · doi:10.24033/asens.1401 |
| [13] | Güler, Osman, Hyperbolic polynomials and interior point methods for convex programming, Math. Oper. Res., 22, 2, 350-377 (1997) · Zbl 0883.90099 · doi:10.1287/moor.22.2.350 |
| [14] | Hartshorne, Robin, Algebraic geometry, 52, xvi+496 p. pp. (1977), Springer · Zbl 0367.14001 |
| [15] | Helton, J. William; Vinnikov, Victor, Linear matrix inequality representation of sets, Commun. Pure Appl. Math., 60, 5, 654-674 (2007) · Zbl 1116.15016 · doi:10.1002/cpa.20155 |
| [16] | Hörmander, Lars, The analysis of linear partial differential operators. II. Differential operators with constant coefficients, viii+392 p. pp. (2005), Springer · Zbl 1062.35004 |
| [17] | Huisman, Johannes, On the geometry of algebraic curves having many real components, Rev. Mat. Complut., 14, 1, 83-92 (2001) · Zbl 1001.14021 |
| [18] | Huisman, Johannes, Non-special divisors on real algebraic curves and embeddings into real projective spaces, Ann. Mat. Pura Appl., 182, 1, 21-35 (2003) · Zbl 1072.14072 · doi:10.1007/s10231-002-0053-1 |
| [19] | Klein, Felix, Gesammelte mathematische Abhandlungen, Über Riemanns Theorie der algebraischen Funktionen und ihrer Integrale, 499-573 (1923), Springer · JFM 49.0011.02 |
| [20] | Kummer, Mario; Shamovich, Eli, Real fibered morphisms and Ulrich sheaves, J. Algebr. Geom., 29, 1, 167-198 (2020) · Zbl 1470.14082 · doi:10.1090/jag/735 |
| [21] | Marcus, Adam W.; Spielman, Daniel A.; Srivastava, Nikhil, Interlacing families I: Bipartite Ramanujan graphs of all degrees, Ann. Math., 182, 1, 307-325 (2015) · Zbl 1316.05066 · doi:10.4007/annals.2015.182.1.7 |
| [22] | Marcus, Adam W.; Spielman, Daniel A.; Srivastava, Nikhil, Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem, Ann. Math., 182, 1, 327-350 (2015) · Zbl 1332.46056 · doi:10.4007/annals.2015.182.1.8 |
| [23] | Mikhalkin, Grigory, Real algebraic curves, the moment map and amoebas, Ann. Math., 151, 1, 309-326 (2000) · Zbl 1073.14555 · doi:10.2307/121119 |
| [24] | Mikhalkin, Grigory; Orevkov, Stepan, Maximally writhed real algebraic links, Invent. Math., 216, 1, 125-152 (2019) · Zbl 1426.57017 · doi:10.1007/s00222-018-0844-7 |
| [25] | Namba, Makoto, Families of meromorphic functions on compact Riemann surfaces, 767, xii+284 p. pp. (1979), Springer · Zbl 0417.32008 |
| [26] | Passare, Mikael; Risler, Jean-Jacques, On the curvature of the real amoeba, Proceedings of the Gökova Geometry-Topology Conference 2010, 129-134 (2011), International Press · Zbl 1243.14049 |
| [27] | Plaumann, Daniel; Sturmfels, Bernd; Vinzant, Cynthia, Quartic curves and their bitangents, J. Symb. Comput., 46, 6, 712-733 (2011) · Zbl 1214.14049 · doi:10.1016/j.jsc.2011.01.007 |
| [28] | Prasolov, Victor V., Elements of homology theory, 81, x+418 p. pp. (2007), American Mathematical Society · Zbl 1120.55001 |
| [29] | Renegar, James, Hyperbolic programs, and their derivative relaxations, Found. Comput. Math., 6, 1, 59-79 (2006) · Zbl 1130.90363 · doi:10.1007/s10208-004-0136-z |
| [30] | Rokhlin, Vladimir A., Complex topological characteristics of real algebraic curves, Usp. Mat. Nauk, 33, 5-203, 77-89, 237 (1978) · Zbl 0437.14013 |
| [31] | Scheiderer, Claus, Sums of squares of regular functions on real algebraic varieties, Trans. Am. Math. Soc., 352, 3, 1039-1069 (2000) · Zbl 0941.14024 · doi:10.1090/S0002-9947-99-02522-2 |
| [32] | Shamovich, Eli; Vinnikov, Victor, Livsic-type determinantal representations and hyperbolicity, Adv. Math., 329, 487-522 (2018) · Zbl 1391.32009 · doi:10.1016/j.aim.2016.06.028 |
| [33] | Touzé, Séverine Fiedler-Le, Totally real pencils of cubics with respect to sextics · Zbl 1192.41001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
