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{\huge Symplectic geometry seminar}
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{ Monday, 2 March 2009, 2:10pm \\
Bahen 6183 }

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{\Large {\bf PINAKI MONDAL}} \\
University of Toronto 

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{\large Projective completions of affine varieties via degree-like functions}

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Given a `degree like' function $\delta$, i.e. a map of the coordinate ring $A$ of an affine variety $X$ to integers such that
$\delta(f+g) \leq \max\{\delta(f), \delta(g)\}$ (with strict inequality
implying $\delta(f) = \delta(g)$) and $\delta(fg) \leq \delta(f) +
\delta(g)$, one can associate to it a projective completion of
$X^\delta$ of $X$. We show that the ideal $I_\infty$ of the hypersurface
of `points at infinity' is radical iff $\delta$ is the maximum of
finitely many semidegrees (i.e. degree like functions $\delta'$ such
that $\delta'(fg) = \delta'(f) + \delta'(g)$ for all $f,g$), which are
then in a 1-1 correspondence with the prime components of $I_{\infty}$.

Given a projective completion $Z$ of $X$ determined by a degree like
function $\delta$, we define a `normalized' degree like function $\bar
\delta$ which is a maximum of finitely many semidegrees. When $X$ is
normal, the corresponding completion is isomorphic to the normalization of
$Z$. The construction of $\bar \delta$ from $\delta$ generalizes the
construction of the normal toric completion $X_P$ of $(C^*)^n$ determined
by a convex integral polytope $P$ from the toric variety determined by an
arbitrary finite subset $S$ of integral points in $P$ whose convex hull is
$P$. I will describe this construction and the properties of toric
completions these are so far known to preserve.

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