Quasismooth hypersurfaces of toric varieties

Michela Artebani
Universidad de Concepción
Abstract:
D. Cox proved that any normal projective toric variety
\[X\]
is the quotient of an open subset
\[\hat X\]
of an affine space for the action of an abelian group 
\[G\]
(for example
\[\mathbb P^n\]
is the quotient of
\[\mathbb A^n\backslash\{0\}\]
for a
\[\mathbb C^*-\]
action) [1]. A hypersurface
\[Y\]
of 
\[X\]
defined as the zero set of a
\[G\]
-homogeneous polynomial
\[f\]
is called quasismooth if the singular locus of 
\[f\]
has empty intersection with
\[\hat X\]
. Quasismooth hypersurfaces are not smooth in general but they inherit the singularities of the ambient space. In this talk I will give a combinatorial characterization of quasismoothness in terms of the Newton polytope of the hypersurface. This is joint work in progress with P. Comparin and R. Guilbot.
References
[1] David A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17–50.