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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Irrational connected sums and the topology of algebraic surfaces

Author: Richard Mandelbaum
Journal: Trans. Amer. Math. Soc. 247 (1979), 137-156
MSC: Primary 57R15; Secondary 14J99
MathSciNet review: 517689
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Abstract: Suppose W is an irreducible nonsingular projective algebraic 3-fold and V a nonsingular hypersurface section of W. Denote by $ {V_m}$ a nonsingular element of $ \left\vert {mV} \right\vert$. Let $ {V_1}$, $ {V_m}$, $ {V_{m\, + \,1}}$ be generic elements of $ \left\vert V \right\vert$, $ \left\vert {mV} \right\vert$, $ \left\vert {(m\, + \,1)V} \right\vert$ respectively such that they have normal crossing in W. Let $ {S_{1m}}\, = \,{V_1}\, \cap \,{V_m}$ and $ C\, = \,{V_1}\, \cap \,{V_m}\, \cap \,{V_{m + 1}}$. Then $ {S_{1m}}$ is a nonsingular curve of genus $ {g_m}$ and C is a collection of $ N\, = \,m\left( {m + 1} \right)V_1^3$ points on $ {S_{1m}}$. By [MM2] we find that $ ( \ast )\,{V_{m\, + \,1}}$ is diffeomorphic to $ \overline {{V_m}\, - \,T({S_{1m}})} \,{ \cup _\eta }\,\overline {{V_1}'\, - \,T({S_{1m}}')} $, where $ T\left( {{S_{1m}}} \right)$ is a tubular neighborhood of $ {S_{1m}}$ in $ {V_m}$, $ {V_1}'$ is $ {V_1}$ blown up along C, $ {S_{1m}}'$ is the strict image of $ {S_{1m}}$ in $ {V_1}'$, $ T({S_{1m}}')$ is a tubular neighborhood of $ {S_{1m}}'$ in $ {V_1}'$ and $ \eta :\,\partial T\left( {{S_{1m}}} \right) \to \partial T({S_m}')$ is a bundle diffeomorphism.

Now $ {V_1}'$ is well known to be diffeomorphic to $ {V_1}\, \char93 \,N\left( { - C{P^2}} \right)$ (the connected sum of $ {V_1}$ and N copies of $ C{P^2}$ with opposite orientation from the usual). Thus in order to be able to inductively reduce questions about the structure of $ {V_m}$ to ones about $ {V_1}$ we must simplify the ``irrational sum'' $ ( \ast )$ above.

The general question we can ask is then the following: Suppose $ {M_1}$ and $ {M_2}$ are compact smooth 4-manifolds and K is a connected q-complex embedded in $ {M_i}$. Let $ {T_i}$ be a regular neighborhood of K in $ {M_i}$ and let $ \eta :\,\partial {T_1}\, \to \,\partial {T_2}$ be a diffeomorphism: Set $ V\, = \,\overline {{M_1}\, - \,{T_1}} \, \cup \,\overline {{M_2}\, - \,{T_2}} $. How can the topology of V be described more simply in terms of those of $ {M_1}$ and $ {M_2}$.

In this paper we show how surgery can be used to simplify the structure of V in the case $ q\, = \,1,\,2$ and indicate some applications to the topology of algebraic surfaces.

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Article copyright: © Copyright 1979 American Mathematical Society