Enumeration of complex and real surfaces via tropical geometry
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AbstractWe prove a correspondence theorem for singular tropical surfaces in ℝ3, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we develop a three-dimensional version of Mikhalkin’s lattice path algorithm that enumerates singular tropical surfaces passing through an appropriate configuration of points in ℝ3. As application we show that there are pencils of real surfaces of degreedin ℙ3containing at least (3/2)d3+O(d2) singular surfaces, which is asymptotically comparable to the number 4(d− 1)3of all complex singular surfaces in the pencil. Our result relies on the classification of singular tropical surfaces [12].
1998 ◽
Vol 102
(3)
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pp. 748-760
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2010 ◽
Vol 147
(1)
◽
pp. 161-187
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2014 ◽
Vol 35
(7)
◽
pp. 2242-2268
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Keyword(s):