Abstract
In this paper the birational surgeries that are necessary to regularize the “addition” of pairs of points to an algebraic surface are described. The pattern of the description is the addition of points to an algebraic surface, which requires the single blow-up of a smooth subvariety, namely, the diagonal in the direct product of the surface by itself. The complete description of the blow-ups leads to a description of the geometry (the ring of algebraic cycles) and the numerical invariants, i.e., the products of the Chern classes of the standard vector bundles on order 4 Hilbert schemes of algebraic surfaces.
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A. S. Tikhomirov, “The variety of complete pairs of zero-dimensional subschemes of an algebraic surface,”Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.],61, No. 6, 153–180 (1997).
W. Fulton, Intersection Theory, Springer-Verlag, Berlin etc., (1984).
I. R. Shafarevich,Basic Algebraic Geometry [in Russian], Vol. 1, Nauka, Moscow (1988).
B. G. Moishezon, “Onn-dimensional compact complex manifolds havingn algebraically independent meromorphic functions I,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],30, No. 1, 133–174 (1966).
J. Harris,Algebraic Geometry. A First Course, Springer, New York (1992).
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Translated fromMatematicheski Zametki, Vol. 65, No. 3, pp. 412–419, March, 1999.
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Tikhomirov, A.S., Troshina, T.L. Birational and numerical geometry of the variety of complete pairs of two-point spaces on an algebraic surface. Math Notes 65, 344–350 (1999). https://doi.org/10.1007/BF02675077
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DOI: https://doi.org/10.1007/BF02675077