de Jong, A. J.; Steenbrink, J. H. M. Proof of a conjecture of W. Veys. (English) Zbl 0857.14019 Indag. Math., New Ser. 6, No. 1, 99-104 (1995). In this article the authors prove a conjecture of W. Veys about the complement of curves in the complex projective plane \(\mathbb{P}^2 (\mathbb{C})\): Let \(C_i\), \(i=1, \dots,n\), be distinct irreducible curves in \(\mathbb{P}^2 (\mathbb{C})\); if the topological Euler characteristic \(e(\mathbb{P}^2 (\mathbb{C}) \backslash \bigcup^n_{i=1} C_i)\geq 0\), then each \(C_i\) is a rational curve. In the first part of the article the authors study a one-parameter family of plane curves. Let \(f:X\to S\) be a proper flat holomorphic mapping from a reduced complex surface \(X\) to the unit disc \(S\), such that the restriction \(X\backslash X_0 \to S\backslash \{0\}\) is a topological fibre bundle. By using perversity of the sheaf complex of nearby cycles \(\psi_f (\mathbb{C}_X)[1]\) and of vanishing cycles \(\varphi_f(\mathbb{C}_X)[1]\) of \(\mathbb{C}_X[2]\) for \(f\), the authors can prove the following inequality between Euler characteristics, which is the main argument to obtain the conjecture.Proposition: Let \((X_t)_{t\in L}\) be a pencil of plane curves. Suppose that the general member \(F\) of this pencil is irreducible. Then \(e(X_t)\geq e(F)\) for all \(t\in T\). If the conjecture was not verified, we could suppose there exist a minimal counterexample with the following properties:(1) \(C_1\) is not a rational curve.(2) \(n\geq 3\), and for \(2\leq i\leq n\) we have \(e(C^\circ_i)=1\), where \(C^\circ_i=C_i \backslash \bigcup_{j\neq i}C_i\).We can define the pencil of curves of degree \(d\) spanned by \(n_1C_1\) and \(n_2C_2\): \[ f:X= \{(x, (\lambda_1: \lambda_2)) \in\mathbb{P}^2 (\mathbb{C}) \times \mathbb{P}^1(\mathbb{C})|\;\lambda_1 F_1(x)^{n_1} + \lambda_2F_2(x)^{n_2}=0\} \to\mathbb{P}^1 (\mathbb{C}) \] where \(F_i\) is the polynomial of degree \(d_i\) defining \(C_i\), and lcm\((d_1,d_2) = d=n_1d_1=n_2d_2\). The general fiber of the pencil \(f:X \to\mathbb{P}^1(\mathbb{C})\) is irreducible. Let \(g:Y \to\mathbb{P}^1 (\mathbb{C})\) be a smooth minimal model for \(f\), as \(C_1\) is not rational, its normalization \(\overline C_1\) occurs as an irreducible component of \(g^{-1} (0)\) and \(H^1(Y_0, \mathbb{Q})\neq 0\). By the proposition and the invariant cycle theorem we deduce \(H^1(Y,\mathbb{Q}) \neq 0\), and we get a contradiction because \(Y\) is birational to \(X\), so is rational. Reviewer: M.Vaquie (Paris) Cited in 1 ReviewCited in 5 Documents MSC: 14H45 Special algebraic curves and curves of low genus 14M20 Rational and unirational varieties 14H10 Families, moduli of curves (algebraic) 14F45 Topological properties in algebraic geometry Keywords:rationality of curves; complement of curves in the complex projective plane; Euler characteristic; one-parameter family of plane curves; perversity of the sheaf complex PDFBibTeX XMLCite \textit{A. J. de Jong} and \textit{J. H. M. Steenbrink}, Indag. Math., New Ser. 6, No. 1, 99--104 (1995; Zbl 0857.14019) Full Text: DOI References: [1] Brylinski, J.-L., ((Co)homologie d’intersection et faisceaux pervers. Sém. Bourbaki. (Co)homologie d’intersection et faisceaux pervers. Sém. Bourbaki, 34e année, 1981/82, exp. 585. (Co)homologie d’intersection et faisceaux pervers. Sém. Bourbaki. (Co)homologie d’intersection et faisceaux pervers. Sém. Bourbaki, 34e année, 1981/82, exp. 585, Astérisque, 92-93 (1982)), 129-157 [2] Beilinson, A. A.; Bernstein, J.; Deligne, P., Faisceaux pervers, (Astérisque, 100 (1982), Soc. Math. de France) · Zbl 0536.14011 [3] Deligne, P., Théorie de Hodge II, Publ. Math. IHES, 40, 5-57 (1971) · Zbl 0219.14007 [4] Hamm, H. A., Lefschetz theorems for singular varieties, (Proc. Symp. Pure Math., 40 (1983)), 547-557, Part I [5] Willem, Veys, Numerical data of resolutions of singularities and Igusa’s local zeta function, (Thesis (1991), Katholieke Universiteit Leuven) · Zbl 0812.14014 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.