Plane curves whose singular points are cusps

Author:
Hisao Yoshihara

Journal:
Proc. Amer. Math. Soc. **103** (1988), 737-740

MSC:
Primary 14H20; Secondary 14H45

DOI:
https://doi.org/10.1090/S0002-9939-1988-0947648-8

MathSciNet review:
947648

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Abstract: Let be an irreducible curve of degree in the complex projective plane. We assume that each singular point is a one place point with multiplicity 2 or 3. Let be the sum of "the Milnor numbers" of the singularities. Then we shall show that . This gives a necessary condition for the existence of such a curve, for example, if is rational, then .

**[1]**F. Hirzebruch,*Some examples of algebraic surfaces*, Contemp. Math., vol. 9, Amer. Math. Soc., Providence, R. I., 1982, pp. 55-71. MR**655974 (83m:14026)****[2]**-,*Singularities of algebraic surfaces and characteristic numbers*, Contemp. Math., vol. 58, Amer. Math. Soc., Providence, R. I., 1986, pp. 141-155. MR**860410 (87j:14057)****[3]**S. Iitaka,*Algebraic geometry, an introduction to birational geometry of algebraic varieties*, Graduate Texts in Math., no. 76, Springer-Verlag, Berlin and New York, 1981. MR**637060 (84j:14001)****[4]**S. Lefschetz,*On the existence of loci with given singularities*, Trans. Amer. Math. Soc.**14**(1913), 23-41. MR**1500934****[5]**Y. Miyaoka,*The maximal number of quotient singularities on surfaces with given numerical invariants*, Math. Ann.**268**(1984), 159-171. MR**744605 (85j:14060)****[6]**I. Wakabayashi,*On the logarithmic Kodaira dimension of the complement of a curve in*, Proc. Japan Acad.**54A**(1978), 157-162. MR**0498590 (58:16683)****[7]**H. Yoshihara,*On plane rational curves*, Proc. Japan Acad.**55A**(1979), 152-155. MR**533711 (80f:14016)****[8]**-,*A note on the existence of some curves*, (to appear).**[9]**O. Zariski,*On the non-existence of curve of order*8*with*16*cusps*, Amer. J. Math.**53**(1931), 309-318. MR**1506819**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0947648-8

Keywords:
Plane curve,
cusp,
Milnor number

Article copyright:
© Copyright 1988
American Mathematical Society