On the incidence cycles of a curve: Some geometric interpretations (Q1314931)

We note that the incidence cycles of a seminormal curve \(X\) intervene in the calculation of the arithmetic genus \(p_ a(X)\), of the algebraic fundamental group \(\pi_ 1^{\text{alg}}(X)\) and of the Picard group \(\text{Pic}(X)\) of \(X\). Really we do not consider only seminormal curves, but more generally varieties obtained from a smooth variety by glueing a finite set of points. Let \(X\) be a connected variety (by a variety we mean a reduced quasi- projective scheme over an algebraically closed field \(k\)) of pure dimension \(r\) whose singular locus \(\text{Sing}(X)\) consists of a finite set of points \(P_ 1,\dots,P_ m\), such that the normalization \(\overline{X}\) of \(X\) is a smooth variety having \(n\) connected components, every one of them of dimension \(r\), \(\overline{X}_ 1,\dots,\overline{X}_ n\) and the normalisation morphism \(\pi: \overline{X} \to X\) is the composition of a finite number of glueing morphisms, that is every singular point of \(X\) is the glueing of a finite number of points of \(\overline{X}\). Let \(M\) be the number of the points of \(\pi^{-1}(\text{Sing}(X))\); we define \(\nu(X) = M - m - n+1\). We associate to \(X\) the graph \(\Gamma\) whose vertices are \(P_ 1,\dots,P_ m,X_ 1,\dots,X_ n\) and whose edges represent the \(M\) branches of \(X\) in this way: if \(x_ r\) is a branch over \(P_ i\) and \(x_ r \in \overline{X}_ j\), an edge joining \(P_ i\) and \(X_ j\) is constructed. Any cycle of the graph \(\Gamma\) associated to \(X\) is said to be an incidence cycle of \(X\). \(\nu(X)\) is the number of the independent incidence cycles of \(X\) [see the author, J. Pure Appl. Algebra 35, 77-83 (1985; Zbl 0579.13003)]. We prove the following results: (1) If \(X\) is projective, we have \(p_ a(X) = p_ a(\overline{X}_ 1)+\dots + p_ a(\overline{X}_ n) + (-1)^{r-1}\nu(X)\). (2) We have \(\pi_ 1^{\text{alg}}(X) \cong (\pi_ 1^{\text{alg}}(\overline{X}_ 1) * \dots * \pi_ 1^{\text{alg}}(\overline{X}_ n) * L_{\nu(X)})^ \wedge\), where \(L_ \nu\) denotes the free group with \(\nu\) generators, \(*\) denotes the free product of groups and \(\wedge\) denotes the completion of the group. (3) We have \(\text{Pic}(X)\cong \text{Pic}(\overline{X}_ 1) \oplus \dots \oplus \text{Pic}(\overline{X}_ n)\oplus \nu(X)k^*\), where \(k^*\) is the multiplicative group \(k-\{0\}\) and \(\nu k^*\) denotes the direct sum of \(\nu\) copies of \(k^*\).