# Abelsche Funktionen und algebraische Geometrie MAg by Conforto F.

By Conforto F.

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3. Let X be a (not necessarily affine) toric variety and X E Y(T), a one-parameter subgroup. If X extends to a morphism from C to X , then the limit X(0) := s+o lim X(s) (22) exists in X . We only use the symbol X(0) in this situation. 4, we have seen that the inclusion r 5 u of a face induces an open embedding X , Q X u of affine toric varieties. 2. Let d u be an inclusion of N-cones. Then the induced morphism Xu,--+ X u is an open embedding (if and) only if u‘ 5 u. Proof. We may interpret U := X u , as an open subset of X := X u .

1. A two-dimensional N-cone admits a unique minimal resolution; in particular, every resolution is a refinement of the minimal one. Proof. We may assume that N = No. 4. The boundary of this polyhedron consists of two unbounded half-lines, each one included in an edge of 0,and finitely many bounded line segments. It thus contains only finitely many primitive lattice points, say YO,. . ,Y‘+’ E N in clockwise order. We set pi := ray(vi), ( ~ := i ei+ei+l, and A := {T ; T 5 ( ~ for i some i , 0 5 i 5 r ) 49 ..

2. 1 (2). 3, for any dimension n, the group G has a system of at most n-1 generators. Non-cyclic groups actually occur already in dimension n = 3. An example is furnished by (T = cone(wl,v2,~3) with w1 = f l , w2 = f l + 2 f 2 , and w3 = f l +2f3. To close this subsection, we have to add at least some remarks on the nonsimplicial case. As usual, we may restrict to full-dimensional cones. 4. The structure of general toric singularities is considerably more complicated. There is a close relation with polytopes spanned by lattice vectors (called lattice polytopes for short): Intersecting a given n-cone o with the affine hyperplanes H p := {u E N R ; (w,p) = 1 ) for any p E ((T')" n M (on a suitable integral "level" 1 > 0) associates to (T a family of (n-1)-dimensional lattice polytopes with fixed combinatorial type.