Abstract
The structures of affine varieties of dimension greater than two can be explored with the help of fibrations by the affine line or plane and quotient morphisms by \(G_a\)-actions. We consider \(G_a\)-actions on affine threefolds and discuss the structure and the singularities of the quotient surface as well as the singular fibers of the quotient morphism.
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Notes
Mumford [27] proved that if a small nice punctured neighborhood of a normal surface singular point Q is simply connected, then Q is a smooth point of the surface. This implies by an easy covering space argument that if \(\varphi : ({\mathbb C}^2,O) \rightarrow (S,Q)\) is a finite surjective complex analytic map of germs with S normal, then the fundamental group G of the germ \(S{\setminus }\{Q\}\) is a finite group and the germ (S, Q) is isomorphic to the germ \(({\mathbb C}^2,O)/G\), where G acts freely on the germ \(({\mathbb C}^2,O)\). This observation applied to a finite analytic map of the germs \(f : (U,P) \rightarrow (Y,Q)\) implies that Q is a quotient singular point, where P is a smooth point of the fiber F, U is a transversal section of F at P and \(Q=f(P)\). See Brieskorn [4, Satz 2.8].
To be more accurate, we have to consider the case where the surfaces \(Y_p\) and \(Y_{p'}\) contain separately disjoint fiber components of the same singular (degenerate) fiber of \(q_1\) resulting the non-empty intersection of the images \(B_p\) and \(B_{p'}\) of \(Y_p\) and \(Y_{p'}\) although \(Y_p\cap Y_{p'}=\emptyset \). By [15, Theorem 1.11], \(q_1\) has equi-dimension one since X is factorial. Since one singular fiber contains finitely many irreducible components, this situation does not occur if we take general points \(p, p'\) in Z. Namely, if \(B_p\cap B_{p'}\ne \emptyset \), then \(Y_p\cap Y_{p'} \ne \emptyset \) and hence \(G_2P\cap G_2P' \ne \emptyset \) for \(P\in C_p\) and \(P'\in C_{p'}\).
A normal algebraic surface with at worst quotient singularities is called a logarithmeic surface.
Let \((V,D+\Gamma )\) be a pair of a smooth projective surface and an effective reduced divisor \(D+\Gamma \) with simple normal crossings. Assume that \(D\cap \Gamma =\emptyset \) and \(\Gamma \) is the exceptional locus of the minimal resolution of a rational double point. Hence each irreducible component of \(\Gamma \) is a smooth rational curve with self-intersection \(-2\), and the intersection form of \(\Gamma \) is negative definite. If \(|n(K_V+D+\Gamma )| \ne \emptyset \) for \(n > 0\), then \(n\Gamma \) is contained in the fixed part. In fact, let \(\Gamma _1\) be the maximal effective divisor such that \(\Gamma _1\) is supported by the irreducible components of \(\Gamma \) and \(\Gamma _1\) is contained in the fixed part of \(|n(K_V+D+\Gamma )|\). Let A be a general member of \(|n(K_V+D+\Gamma )|\). Then \(A-\Gamma _1\) has no component of \(\Gamma \). Suppose that \(n\Gamma > \Gamma _1\), and write \(n\Gamma =\Gamma _1+\Gamma _2\). Then \((A-\Gamma _1)\cdot \Gamma _2 \ge 0\). Meanwhile, \((A-\Gamma _1)\cdot \Gamma _2=(n(K_V+D+\Gamma )-\Gamma _1)\cdot \Gamma _2=(n(K_V+D)+\Gamma _2)\cdot \Gamma _2=\Gamma _2^2 < 0\), which is a contradiction. So, \(\Gamma _1 \ge n\Gamma \). This implies that \(\overline{\kappa }(V{\setminus }(D+\Gamma ))=\overline{\kappa }(V{\setminus } D)\). The case of non-minimal resolution is reduced to the case of minimal resolution.
If \(n=4\), we assume that Y exists, i.e., the ring of invariants is an affine domain over k. If \(n =2, 3\), the ring of invariants is an affine domain over k by Zariski’s Finiteness Theorem (see [11, p.147]).
see Remark 4.5 below.
Suppose that \(\Delta _{(F,G)}=AD\) for a derivation D. There exists then an element \(\xi \in k[x,y,z]\) such that \(\Delta _{(F,G)}(\xi )\) is a nonzero element of \(\mathrm{Ker}\,\Delta _{(F,G)}\). Then a factor A of an element of \(\mathrm{Ker}\,\Delta _{(F,G)}\) is also in the same kernel.
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Communicated by Ngaiming Mok.
M. Koras passed away on September 15, 2017. The first author was supported by Dr. Raja Ramanna Fellowship of the Department of Atomic Energy of India at, the Indian Institute of Technology, Bombay. The second author was supported by the Polish National Center of Science, Grant No. 2013/11/B/ST1/02977. The third and fourth authors were respectively supported by Grant-in-Aid for Scientific Research (C), No. 15K04831 and (B), No. 24340006, JSPS. The fifth author was supported by NSERC, Canada.
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Gurjar, R.V., Koras, M., Masuda, K. et al. Affine threefolds admitting \(G_a\)-actions. Math. Ann. 373, 1211–1236 (2019). https://doi.org/10.1007/s00208-017-1622-3
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DOI: https://doi.org/10.1007/s00208-017-1622-3