Joins and intersections
The embedded join of two irreducible projective varieties \(X\) and \(Y\) in \(\mathbb{P}^ n=\mathbb{P}^ n_ k\) \((k\) being an arbitrary field) is the union of all lines \(xy\) joining a point \(x\) of \(X\) and a point \(y\) of \(Y\). We shall give it the reduced induced subscheme structure and we shall denote it by \(XY\). By \textit{B. Ådlandsvik} [Math. Scand. 61, No. 2, 213-222 (1987; Zbl 0657.14034)] it is known that \[ \dim X+\dim Y-\dim X\cap Y\leq\dim XY\leq\dim X+\dim Y+1. \] Moreover we have \(\deg XY\leq\deg X\cdot\deg Y\). --- The authors improve the above results proving, among other things, that \((\circ)\) The embedded join has minimal dimension: \(\dim XY=\dim X+\dim Y-\dim X\cap Y\), if and only if \(\deg X\cdot\deg Y=\sum_{C irr}e_ C\deg C\). In this case \(\deg XY\leq e_ C\) for all \(C\) and moreover all irreducible components have the same dimension. In the last formula the summation is extended to the irreducible components \(C\) of \(X\cap Y\) and \(e_ C=(e_{X\cap Y}X\times Y)_ C\) is the natural multiplicity of Samuel. \((\circ\circ)\) If \(\dim XY>\dim X+\dim Y-\dim X\cap Y\), then \[ \deg X\cdot\deg Y\geq\sum_{C irr}e_ C\deg C+\dim XY+\dim X\cap Y-\dim X- \dim Y+\deg XY-1. \] From the above results the authors derive an unpublished result due to \textit{R. Lazarsfeld}: if \(XY\) is non- degenerated, then \(\deg X\cdot\deg Y\geq\sum\deg C+e\), where \(C\) runs through the irreducible components of \(X\cap Y\) and \(e=n+\dim X\cap Y- \dim X-\dim Y\) is the excess dimension of the intersection of \(X\) and \(Y\).
scientific article; zbMATH DE number 59943
scientific article