On the geometry of the sequence of infinitely near points (Q1814167)

Let \(X\) be a smooth variety over an algebraically closed field \(k\) of characteristic zero, \(P_ 0\in X\) a closed point and \(E_ 1\) the exceptional divisor of the blowing-up of \(X\) with center \(P_ 0\). A closed point \(P_ 1\in E_ 1\) is said to be an infinitely near point (shortly i.n.p.) of \(P_ 0\). A sequence of i.n.p. is defined obviously by recurrence. This paper studies some invariants of the curves \(C\) passing through these i.n.p. as e.g. the multiplicity sequence \(e(C_ i)\) of \(C\), \(C_ i\) being the \(i\)-th quadratic transform of \(C\), the Arf dimensions of \(C_ i\), the semigroups of the saturated rings (cf. Zariski or Campillo saturation) of \(A_ i\), where \(C_ i=\text{Spec}(A_ i)\) and the semigroups of saturated rings of \(A_ i\) relative to some exceptional divisors of the chain of blowing up's.