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Izv. RAN. Ser. Mat., 1993, Volume 57, Issue 1, Pages 76–101 (Mi izv888)  

This article is cited in 9 scientific papers (total in 9 papers)

Alexander polynomials of plane algebraic curves

Vik. S. Kulikov

Abstract: The author studies the fundamental group of the complement of an algebraic curve $D\subset\mathbf C^2$ defined by an equation $f(x,y)=0$. Let $F\colon X=\mathbf C^2\setminus D\to\mathbf C^*=\mathbf C\setminus\{0\}$ be the morphism defined by the equation $z=f(x,y)$. The main result is that if the generic fiber $Y=F^{-1}(z_0)$ is irreducible, then the kernel of the homomorphism $F_*\colon\pi_1(X)\to\pi_1(\mathbf C^*)$ is a finitely generated group. In particular, if $D$ is an irreducible curve, then the commutator subgroup of $\pi_1(X)$ is finitely generated.
The internal and external Alexander polynomials of a curve $D$ (denoted by $\Delta_{in}(t)$ and $\Delta_{ex}(t)$ respectively) are introduced, and it is shown that the Alexander polynomial $\Delta_1(t)$ of the curve $D$ divides $\Delta_{in}(t)$ and $\Delta_{ex}(t)$ and is a reciprocal polynomial whose roots are roots of unity. Furthermore, if $D$ is an irreducible curve, the Alexander polynomial $\Delta_1(t)$ of the curve $D$ satisfies the condition $\Delta_1(1)=\pm1$. From this it follows that among the roots of the Alexander polynomial $\Delta_1(t)$ of an irreducible curve there are no primitive roots of unity of degree $p^n$, where $p$ is a prime number.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1994, 42:1, 67–89

Bibliographic databases:

Document Type: Article
UDC: 512.7+515.1
MSC: 14E20, 14E22, 14F45, 14J25, 20F34, 57M05
Received: 24.03.1992

Citation: Vik. S. Kulikov, “Alexander polynomials of plane algebraic curves”, Izv. RAN. Ser. Mat., 57:1 (1993), 76–101; Russian Acad. Sci. Izv. Math., 42:1 (1994), 67–89

Citation in format AMSBIB
\by Vik.~S.~Kulikov
\paper Alexander polynomials of plane algebraic curves
\jour Izv. RAN. Ser. Mat.
\yr 1993
\vol 57
\issue 1
\pages 76--101
\jour Russian Acad. Sci. Izv. Math.
\yr 1994
\vol 42
\issue 1
\pages 67--89

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    This publication is cited in the following articles:
    1. Vik. S. Kulikov, “A geometric realization of $C$-groups”, Russian Acad. Sci. Izv. Math., 45:1 (1995), 197–206  mathnet  crossref  mathscinet  zmath  isi
    2. Vik. S. Kulikov, V. S. Kulikov, “On the monodromy and mixed Hodge structure on cohomology of the infinite cyclic covering of the complement to a plane algebraic curve”, Izv. Math., 59:2 (1995), 367–386  mathnet  crossref  mathscinet  zmath  isi
    3. Vik. S. Kulikov, “On plane algebraic curves of positive Albanese dimension”, Izv. Math., 59:6 (1995), 1173–1192  mathnet  crossref  mathscinet  zmath  isi
    4. Vik. S. Kulikov, “On the fundamental groups of complements of toral curves”, Izv. Math., 61:1 (1997), 89–112  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Vik. S. Kulikov, “Finite presentability of the commutator subgroup of the fundamental group of the complement of a plane curve”, Izv. Math., 61:5 (1997), 961–967  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. G.-M. Greuel, Vik. S. Kulikov, “On symplectic coverings of the projective plane”, Izv. Math., 69:4 (2005), 667–701  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. Vik. S. Kulikov, “Alexander polynomials of Hurwitz curves”, Izv. Math., 70:1 (2006), 69–86  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Vik. S. Kulikov, “Hurwitz curves”, Russian Math. Surveys, 62:6 (2007), 1043–1119  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Vik. S. Kulikov, “Alexander modules of irreducible $C$-groups”, Izv. Math., 72:2 (2008), 305–344  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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