β Scribed by John Atwell Moody
No coin nor oath required. For personal study only.
For V a singular affine irreducible variety over a field k, and D D an O O -module of k-linear derivations, we wish to address the question whether V there is a blowup V of V such that the subsheaf of the constant sheaf of rational functions
ΛαΉΌ V Δ©s a locally free coherent sheaf on V. At one extreme is the case D D s O O ΠΈ β¦ for β¦ g D D a fixed derivation. In V this case, the question is equivalent to whether the vector field β¦ lifts to a Γ±onsingular one-dimensional foliation on V. Significant work has been done on the question of reduction of singularities of vector fields in this Ε½ w x . situation see 1 and its sequels .
At the other extreme, assuming V is normal, for any nonsingular blowup
locally free of rank one.
In both these cases, and for general D D, the problem is equivalent to finding an ideal I ; O O so that, once modified in a certain way to produce V Ε½ .
Ε½ . a new ideal J J I , it must happen that as a fractional ideal J J I is a divisor of a power of I:
For purposes of discussion, let's call this the ''strict condition.'' From the previous paragraph, the problem of finding an ideal I satisfying the strict condition for at least some choice of D D is analogous to, but easier than, the problem of resolving the singularities of V. Moreover, understanding either problem well should enlighten the other. 90
π SIMILAR VOLUMES
We study the dynamical behaviour of polynomial hamiltonian planar vector fields. Particularly we analyze the structure of finite and infinite critical points and we obtain the best upper bound of the number of centers and of the number of saddles that a system of this type can exhibit, depending on