Download SIMPLEST SINGULARITY IN NON-ALGEBRAIC

SIMPLEST SINGULARITY IN NON-ALGEBRAIC NORMAL MOISHEZON
SURFACES
A classical question in complex analytic geometry is to understand when a given analytic space is algebraic
(i.e. analytification of an algebraic scheme). A necessary condition for this to hold is that the transcendence
degree of the field of global meromorphic functions must be equal to the dimension of the space, i.e. the
space has to be Moishezon. For dimension 2, it is a classical result that it is also sufficient, provided X is
non-singular (Chow and Kodaira, 1952).
In general it is not clear how to determine algebraicity of normal (singular) Moishezon surfaces and our understanding of non-algebraic Moishezon surfaces, more precisely what prevents them from being algebraic,
remains incomplete ([Sch00] gives a necessary and sufficient for algebraicity, but it is not suitable for computation in a given case). [Mon16, Section 2] gives an example of a non-algebraic normal Moishezon surface
X which has the simplest possible singularity in the following sense: X has only one singular point P , and
the singularity at P
(1) has multiplicity 2 and geometric genus 1,
(2) is almost rational in the sense of [Ném07],
(3) is a Gorenstein hypersurface singularity which is minimally elliptic (in the sense of [Lau77]).
The claim that singularity of X is the simplest possible is based on combining the preceding facts with the
following observations:
• a Moishezon surface whose singularities are rational (i.e. with geometric genus zero) is algebraic
[Art62], and
• minimally elliptic Gorenstein singularities form in a sense the simplest class of non-rational singularities.
−1
−3
−2
−2
−2
L̃i
E2 E4
E5
E3 −2
−2
E10
E11
string of vertices of weight −2
E1 −3
(a) Weighted dual graph of the resolution of singularities of X
(b) z 2 = x5 + xy 5
The weighted dual graph of the resolution of singularity of X at P is of type D9,∗,0 in the classification of
[Lau77] and the self-intersection number of its fundamental divisor is −2. It follows from [Lau77, Table 2]
that the singularity at the origin of z 2 = x5 + xy 5 is also of the same type.
References
[Art62]
Michael Artin. Some numerical criteria for contractability of curves on algebraic surfaces. Amer. J. Math., 84:485–496,
1962. 1
[Lau77] Henry B. Laufer. On minimally elliptic singularities. Amer. J. Math., 99(6):1257–1295, 1977. 1
[Mon16] Pinaki Mondal. Algebraicity of normal analytic compactifications of C2 with one irreducible curve at infinity. Algebra
& Number Theory, 10(8), 2016. 1
[Ném07] András Némethi. Graded roots and singularities. In Singularities in geometry and topology, pages 394–463. World Sci.
Publ., Hackensack, NJ, 2007. 1
[Sch00] Stefan Schröer. On contractible curves on normal surfaces. J. Reine Angew. Math., 524:1–15, 2000. 1
1