Kulikov, Viktor S.; Shustin, Eugenii On rigid plane curves. (English) Zbl 1352.14017 Eur. J. Math. 2, No. 1, 208-226 (2016). Rigid curves (or \(k\)-rigid) curves are those belonging to minimal equisingular families, which are formed by reduced curves and are the union of finitely many orbits (say \(k\)) of the action of the group of projective transformations of the plane Aut\(\;\mathbb{P}^2\). When \(k=1\), they are called strictly rigid. These curves appear in several important problems in Algebraic Geometry.The main goals of the paper are: – To give a complete list of rigid curves of degree \(\leq 4\). – To provide an infinite series of examples of strictly rigid families of irreducible rational curves in the (equisingular) family of reduced irreducible plane curves of degree \(d\), geometric genus equal to zero and certain equisingularity types. – For each \(g \geq 1\), to prove the existence of strictly rigid irreducible plane curves of genus \(g\). – To give some examples of \(2\)-rigid families of irreducible curves and of \(k\)-rigid families with \(k\) irreducible components for each \(k\). Some open questions about this interesting topic are also included in the paper. Reviewer: Carlos Galindo (Castellón) Cited in 2 Documents MSC: 14H10 Families, moduli of curves (algebraic) 14H20 Singularities of curves, local rings 14H50 Plane and space curves Keywords:plane curves; rigid curves; equisingular families of curves PDFBibTeX XMLCite \textit{V. S. Kulikov} and \textit{E. Shustin}, Eur. J. Math. 2, No. 1, 208--226 (2016; Zbl 1352.14017) Full Text: DOI arXiv References: [1] Belyĭ, G.V.: On Galois extensions of a maximal cyclotomic field. Math. USSR-Izv. 14(2), 247-256 (1980) · Zbl 0429.12004 [2] Greuel, G-M; Lossen, C.; Shustin, E.; Catanese, F. (ed.); etal., Equisingular families of projective curves, 171-209 (2006), Berlin · Zbl 1116.14015 [3] Hirzebruch, F., Some examples of algebraic surfaces, No. 9, 55-71 (1981), Providence · Zbl 0487.14009 [4] Kulikov, Vik.S.: A remark on classical pluecker’s formulae (2011). arXiv:1101.5042 · Zbl 1375.14100 [5] Kulikov, Vik.S., Kharlamov, V.M.: On real structures on rigid surfaces. Izv. Math. 66(1), 133-150 (2002) · Zbl 1055.14060 [6] Paranjape, K.H.: A geometric characterization of arithmetic varieties. Proc. Indian Acad. Sci. Math. Sci. 112(3), 383-391 (2002) · Zbl 1054.14029 [7] Shustin, E.I.: Versal deformations in the space of planar curves of fixed degree. Funct. Anal. Appl. 21(1), 82-84 (1987) · Zbl 0627.14023 [8] Wall, C.T.C.: Geometry of quartic curves. Math. Proc. Cambridge Philos. Soc. 117(3), 415-423 (1995) · Zbl 0863.14018 [9] Zaidenberg, M.G., Orevkov, SYu.: On rigid rational cuspidal plane curves. Russian Math. Surveys 51(1), 179-180 (1996) · Zbl 0878.14016 [10] Zariski, O.: Studies in equisingularity I. Equivalent singularities of plane algebroid curves. Amer. J. Math. 87(2), 507-536 (1965) · Zbl 0132.41601 [11] Zariski, O.: Studies in equisingularity II. Equisingularity in codimension 1 (and characteristic zero). Amer. J. Math. 87(4), 972-1006 (1965) · Zbl 0146.42502 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.