## Quasismooth hypersurfaces of toric varieties

Michela Artebani
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.