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

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:

UDC: 512.7+515.1
MSC: 14E20, 14E22, 14F45, 14J25, 20F34, 57M05

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
\Bibitem{Kul93} \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 \mathnet{http://mi.mathnet.ru/izv888} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1220582} \zmath{https://zbmath.org/?q=an:0811.14017} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1994IzMat..42...67K} \transl \jour Russian Acad. Sci. Izv. Math. \yr 1994 \vol 42 \issue 1 \pages 67--89 \crossref{https://doi.org/10.1070/IM1994v042n01ABEH001534} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994NH32100004} 

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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
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
3. Vik. S. Kulikov, “On plane algebraic curves of positive Albanese dimension”, Izv. Math., 59:6 (1995), 1173–1192
4. Vik. S. Kulikov, “On the fundamental groups of complements of toral curves”, Izv. Math., 61:1 (1997), 89–112
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
6. G.-M. Greuel, Vik. S. Kulikov, “On symplectic coverings of the projective plane”, Izv. Math., 69:4 (2005), 667–701
7. Vik. S. Kulikov, “Alexander polynomials of Hurwitz curves”, Izv. Math., 70:1 (2006), 69–86
8. Vik. S. Kulikov, “Hurwitz curves”, Russian Math. Surveys, 62:6 (2007), 1043–1119
9. Vik. S. Kulikov, “Alexander modules of irreducible $C$-groups”, Izv. Math., 72:2 (2008), 305–344
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