Embedded 2-spheres in indefinite 4-manifolds

The main results are as follows. Let xi=p gamma+q(1) delta(1)+...+q(n) delta(n) be an element of H-2(CP2#n<(CP)over bar>CP2), where gamma,delta(1),...,delta(n) are standard generators. Suppose 2 less than or equal to n less than or equal to 9. (1) If \p\,\q(i)\,...,\q(n)\less than or equal to 2 or \p\=\q(i)\ for some i and \q(j)\less than or equal to 2 for j not equal i or \\p\-\q(i)\\=1 for some i, then xi can be represented by a smoothly embedded 2-sphere. (2) If xi is a characteristic homology class, then xi cannot be represented by a smoothly embedded 2-sphere except for xi(2)=16l+1-n, l=0 and 2 less than or equal to n less than or equal to 9; l=-1 and 4 less than or equal to n less than or equal to 9; l=02 and 7 less than or equal to n less than or equal to 9. (3) If xi=d eta for some d is an element of Z, where eta is a primitive ordinary homology class with eta(2)=0, then xi is eqivalent to (d;d,0,...,0) and can be presented by a smoothly embedded 2-sphere.