Takeshi Kawachi's research while affiliated with Tokyo Institute of Technology and other places
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Publications (5)
We introduce some invariants of singularities which represent the anti-freeness of the adjoint linear systems. The invariants
indicate that if either the singularities or the boundaries are worse then the adjoint linear systems are much global generative.
Using these invariants, we prove effective global generation of adjoint linear systems on norm...
We give the new effective criterion for the global generation of the adjoint bundle on normal surfaces with a boundary. We could make the invariant \delta small a bit more on log-terminal singular point, and then we could prove the theorem described in my previous paper "alg-geom/9612018" as a corollary.
This treats the base-point-freeness of the adjoint bundles on normal surfaces with a boundary. This is an extension of the non-relative version of the theorem of Ein-Lazarsfeld-Masek and the theorem of Kawachi-Masek.
We extend Reider's freeness criterion to normal surfaces of characteristic 0. Let Y be a normal surface. Let D be a nef divisor on Y such that K_Y+D is a Cartier divisor. Let x be a point on Y. If x is a base point of |K_Y+D| and D^2>\delta_x (\delta_x is determined by x, \delta_x <= 4) then there exists a non zero effective divisor E on Y passing...
Let $(X,L)$ be an $n$-dimensional polarized variety. Fujita's conjecture says that if $L^n>1$ then the adjoint bundle $K_X+nL$ is spanned and $K_X+(n+1)L$ is very ample. There are some examples such that $K_X+nL$ is not spanned or $K_X+(n+1)L$ is not very ample. These are $(\P^n,\O(1))$, hypersurface $M$ of degree $6$ in weighted projective space $...
Citations
... In §2 we prove several criteria for global generation of linear systems of the form |K Y +⌈M ⌉|, M a Q -divisor on Y such that K Y +⌈M ⌉ is Cartier. This type of results was the original motivation for introducing the invariant δ y (see [8,9,10]). The main interest in results of Reider type for Q -divisors on normal surfaces comes from work related to Fujita's conjecture for (log-) terminal threefolds, cf. [5,11]. ...
... , and this has P d as its base point. Another series of examples is in [Kawachi96]. ...
Reference: Singularities of Pairs
... In §2 we prove several criteria for global generation of linear systems of the form |K Y +⌈M ⌉|, M a Q -divisor on Y such that K Y +⌈M ⌉ is Cartier. This type of results was the original motivation for introducing the invariant δ y (see [8,9,10]). The main interest in results of Reider type for Q -divisors on normal surfaces comes from work related to Fujita's conjecture for (log-) terminal threefolds, cf. [5,11]. ...
