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The index of an algebraic variety. (English) Zbl 1268.13009

For a non-empty scheme \(W\) of finite type over a field \(F\), let \(\delta(W/F)\) denote the greatest common divisor of the degrees of closed points of \(W\). Let \(K\) be the field of fractions of discrete valuation ring \(\mathcal{O}_K\) with residue field \(k\). Let \(X\rightarrow S=\text{Spec}(\mathcal{O}_K)\) be a proper flat morphism, with \(X\) regular and irreducible. Let \(X_K/K\) be the generic fiber of \(X/S\). Write the special fiber \(X_k\) as \(\sum_{i=1}^nr_i\Gamma_i\), where for each \(i=1,\cdots,n\), \(\Gamma_i\) is irreducible of multiplicity \(r_i\) in \(X_k\). The main result of this paper shows that \(\text{gcd}\{r_i\delta(\Gamma_i^{\mathrm{reg}}/k)\}\) divides \(\delta(X_K/K)\), and that when \(\mathcal{O}_K\) is Henselian, they are equal. This result answers positively a question of P. L. Clark posed in [Manuscr. Math. 124, No. 4, 411-426 (2007; Zbl 1222.11078), Conj. 16]. The authors give two proofs of this result, using two kinds of moving lemmas.

MSC:

13H15 Multiplicity theory and related topics
14C25 Algebraic cycles
14D06 Fibrations, degenerations in algebraic geometry
14D10 Arithmetic ground fields (finite, local, global) and families or fibrations
14G05 Rational points
14G20 Local ground fields in algebraic geometry

Citations:

Zbl 1222.11078
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References:

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