The index of an algebraic variety
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 \textit{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.
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