Proposition 4.2 [014L] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
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Proposition 4.2
On W W
(1) H 3 = 5 H^{3}=5 , H ⋅ E i j l ⋅ E i j l + 1 = 1 H\cdot E^{l}_{ij}\cdot E^{l+1}_{ij}=1 , H ⋅ ( E i j l ) 2 = − 2 H\cdot(E^{l}_{ij})^{2}=-2 , H 2 ⋅ L i = 1 H^{2}\cdot L_{i}=1 ,
H ⋅ L i ⋅ E i j 1 = 1 H\cdot L_{i}\cdot E^{1}_{ij}=1 ,
H ⋅ L i 2 = − 3 H\cdot L_{i}^{2}=-3 .
(2)
L i ⋅ ( E i j 1 ) 2 = 1 , L i 2 ⋅ E i j 1 = − 3 , E i j 1 ⋅ ( E i j 2 ) 2 = 0 , ( E i j 1 ) 2 ⋅ E i j 2 = − 2 , E i j 2 ⋅ ( E i j 3 ) 2 = − 1 , ( E i j 2 ) 2 ⋅ E i j 3 = − 1 , E i j 3 ⋅ ( E i j 4 ) 2 = − 2 , ( E i j 3 ) 2 ⋅ E i j 4 = 0 , \eqalign{L_{i}\cdot(E^{1}_{ij})^{2}=1,\quad&L_{i}^{2}\cdot E^{1}_{ij}=-3,\cr E^{1}_{ij}\cdot(E^{2}_{ij})^{2}=0,\quad&(E^{1}_{ij})^{2}\cdot E^{2}_{ij}=-2,\cr E^{2}_{ij}\cdot(E^{3}_{ij})^{2}=-1,\quad&(E^{2}_{ij})^{2}\cdot E^{3}_{ij}=-1,\cr E^{3}_{ij}\cdot(E^{4}_{ij})^{2}=-2,\quad&(E^{3}_{ij})^{2}\cdot E^{4}_{ij}=0,\cr}
and L i 3 = 9 L_{i}^{3}=9 ,
( E i j l ) 3 = 5 (E^{l}_{ij})^{3}=5 .
(3) Fixing i i , j j and k k , let
{ D 1 , … , D 21 } = { E i j k 1 , … , E i j k 6 , E i j 1 , … , E j k 4 , L i , L j , L k } \{D_{1},\ldots,D_{21}\}=\{E^{1}_{ijk},\ldots,E^{6}_{ijk},E^{1}_{ij},\ldots,E^{4}_{jk},L_{i},L_{j},L_{k}\}
be the 21 divisors corresponding to the 21 vertices in the triangulation
of Figure 4.6. Then D i 2 ⋅ D j = − 1 D_{i}^{2}\cdot D_{j}=-1
if there is an edge ⟨ i , j ⟩ \langle i,j\rangle joining vertex i i
and vertex j j and ⟨ i , j ⟩ \langle i,j\rangle is not an exterior
edge.
Also D i ⋅ D j ⋅ D k = 1 D_{i}\cdot D_{j}\cdot D_{k}=1 if vertices i , j , k i,j,k are the vertices of a 2-simplex in Figure 4.6,
and D i 3 = 6 D_{i}^{3}=6 if D i D_{i} corresponds to an interior vertex.
(4) All other intersection numbers involving H H , the L i L_{i} ’s,
the E i j l E^{l}_{ij} ’s and the E i j k l E^{l}_{ijk} ’s are zero.
(5) If c 2 ∈ H 4 ( W , 𝐙 ) c_{2}\in H^{4}(W,{\bf Z})
is the second Chern class, then
H . c 2 = 50 , E i j l ⋅ c 2 = 2 , E i j k l ⋅ c 2 = 0 , L i ⋅ c 2 = − 6 . \eqalign{H.c_{2}&=50,\cr E^{l}_{ij}\cdot c_{2}&=2,\cr E^{l}_{ijk}\cdot c_{2}&=0,\cr L_{i}\cdot c_{2}&=-6.\cr}