Quasismooth hypersurfaces of toric varieties

Michela Artebani
Universidad de Concepción
D. Cox proved that any normal projective toric variety
is the quotient of an open subset
\[\hat X\]
of an affine space for the action of an abelian group 
(for example
\[\mathbb P^n\]
is the quotient of
\[\mathbb A^n\backslash\{0\}\]
for a
\[\mathbb C^*-\]
action) [1]. A hypersurface
defined as the zero set of a
-homogeneous polynomial
is called quasismooth if the singular locus of 
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.
[1] David A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17–50.