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Special effect varieties in higher dimension. (English) Zbl 1093.14009

The subject of this work are linear systems in \({\mathbb P}^n\) of type \({\mathcal L} = {\mathcal L}_{n,d}(-\sum _{i=1}^h m_iP_i)\), \(P_i\in {\mathbb P}^n\), \(m_i\in {\mathbb N}\); which are given by hypersurfaces of degree \(d\) passing through \(h\) generic points \(P_i\) with multiplicities \(\geq m_i\).
Even for \(n=2\), the dimension of \({\mathcal L}\) is not known in general, but several equivalent conjectures (the first due to B. Segre) state that the dimension is the expected one except in the cases when the linear system contains a multiple fixed component which is a \((-1)\)-curve. In a previous paper, the author proposed yet other two equivalent forms for such conjectures, stating that \({\mathcal L}_{2,d}(-\sum _{i=1}^h m_iP_i)\) is special (i.e. it does not have the expected dimension), if and only if it is “numerically special” or “cohomologically special”, where the main interest for those two concepts is that they could be generalized to \(n\geq 3\), where no general conjecture for the dimension of \({\mathcal L}\) is known.
We say that \({\mathcal L}\) is numerically special if it exists an \(\alpha\)-special effect variety \(Y\) for \({\mathcal L}\), i.e. an irreducible variety \(Y\) such that \({\mathcal L}-\alpha Y\) has positive virtual dimension greater than \({\mathcal L}\) (a few more conditions are required if \(\dim Y = n-1\)). We say that \({\mathcal L}\) is cohomologically special, instead, if it exists a \(h^1\)-special effect variety \(Y\), i.e. an irreducible \(Y\) such that \({\mathcal L}- Y\) has positive dimension, \(h^0({\mathcal L}|_Y)=0\) and \(h^1({\mathcal L}|_Y)>h^2({\mathcal L}-Y)\) (an example showing that the two definitions do not coincide is given).
The author conjectures that for \({\mathcal L}\) to be special is equivalent to being numerically special and also to being cohomologically special. In the case \(n=2\), \(\alpha\)-special effect varieties and \(h^1\)-special effect varieties are actually \((-1)\)-curves, while for \(n\geq 3\) examples are given using rational normal curves, hypersurfaces or linear spaces, and those covers most known examples of special \({\mathcal L}\)’s.
Moreover, the case of linear systems in a product \({\mathbb P}^{n_1}\times {\mathbb P}^{n_2}\times\dots\times {\mathbb P}^{n_t}\) is studied and examples (known and new) are given of systems whose speciality is due to special effect varieties.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14N05 Projective techniques in algebraic geometry
14H20 Singularities of curves, local rings
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