Proof of a conjecture of W. Veys.

*(English)*Zbl 0857.14019In 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.

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)

##### 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
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\textit{A. J. de Jong} and \textit{J. H. M. Steenbrink}, Indag. Math., New Ser. 6, No. 1, 99--104 (1995; Zbl 0857.14019)

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##### References:

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