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An effective Bertini theorem over finite fields. (English) Zbl 1070.14524
Summary: Let \(p\) be a prime, \(q\) a power of \(p\) and \(K\) the algebraic closure of the finite field \(\mathbb{F}_p\). Let \(X\subset\mathbb{P}^N(K)\) be an irreducible variety. Set \(n=\dim(X)\) and \(d=\deg(X)\). Here we prove that if \(q\geq d(d-1)^n\) there is a hyperplane \(H\) of \(\mathbb{P}^N(K)\) defined over \(\mathbb{F}_q\) and transversal to \(X\).

MSC:
14N05 Projective techniques in algebraic geometry
14G15 Finite ground fields in algebraic geometry
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