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**An asymptotic vanishing theorem for generic unions of multiple points.**
*(English)*
Zbl 0973.14026

The main result of this paper can be summarized as follows: “Let \(X\) be a projective, reduced and irreducible scheme, \(L\) an ample line bundle on \(X\) and \(m\geq 0\) an integer. Then there exists an integer \(l\) such that any generic union \(Z\) of fat points of multiplicity at most \(m\) and of total degree (i.e. length) at least \(l\) is such that for all \(d\) the maps: \(H^0(X,L^d)\rightarrow H^0(X,{\mathcal O}_Z\otimes L^d))\) have maximal rank.”

In the case \(X={\mathbb{P}}^n\), this can be expressed by saying that for any scheme of generic “fat points” (infinitesimal neighborhoods) \(Z=(P_1,\dots ,P_s;m_1,\dots ,m_s) \subset {\mathbb{P}}^n\), \(m_1\geq m_2\dots \geq m_s\), there exists an integer \(l(m_1)\) such that for \(\deg Z \geq l(m_1)\), the Hilbert function \(H(z,d)\) of \(Z\) is “as expected”, for all \(d\).

The result is a consequence of a delicate refinement of the well-known “Horace method” which allows to “slice up” a fat point with a divisor in an unusual way, intersecting “the wrong infinitesimal neighborhood”, let’s say. This idea is quite surprising, and its proof is quite technical, but it is very interesting, and hopefully could be applied in a number of similar situations to get results (Hilbert function, resolution) about ideals of fat points.

In the case \(X={\mathbb{P}}^n\), this can be expressed by saying that for any scheme of generic “fat points” (infinitesimal neighborhoods) \(Z=(P_1,\dots ,P_s;m_1,\dots ,m_s) \subset {\mathbb{P}}^n\), \(m_1\geq m_2\dots \geq m_s\), there exists an integer \(l(m_1)\) such that for \(\deg Z \geq l(m_1)\), the Hilbert function \(H(z,d)\) of \(Z\) is “as expected”, for all \(d\).

The result is a consequence of a delicate refinement of the well-known “Horace method” which allows to “slice up” a fat point with a divisor in an unusual way, intersecting “the wrong infinitesimal neighborhood”, let’s say. This idea is quite surprising, and its proof is quite technical, but it is very interesting, and hopefully could be applied in a number of similar situations to get results (Hilbert function, resolution) about ideals of fat points.

Reviewer: Alessandro Gimigliano (Bologna)