9 research outputs found
A Jacobian module for disentanglements and applications to Mond's conjecture
Given a germ of holomorphic map from to ,
we define a module whose dimension over is an upper bound
for the -codimension of , with equality if is weighted
homogeneous. We also define a relative version of the module, for
unfoldings of . The main result is that if are nice
dimensions, then the dimension of over is an upper bound of
the image Milnor number of , with equality if and only if the relative
module is Cohen-Macaulay for some stable unfolding . In particular,
if is Cohen-Macaulay, then we have Mond's conjecture for .
Furthermore, if is quasi-homogeneous, then Mond's conjecture for is
equivalent to the fact that is Cohen-Macaulay. Finally, we observe
that to prove Mond's conjecture, it suffices to prove it in a suitable family
of examples.Comment: 19 page
A jacobian module for disentanglements and applications to Mond's conjecture
Let be a germ whose image is given by . We define an -module with the property that -, with equality if
is weighted homogeneous.
We also define a relative version for unfoldings , in such a way that specialises to when specialises to . The main result is that if are
nice dimensions, then , with equality if and only if is Cohen-Macaulay, for some stable unfolding . Here, denotes the image
Milnor number of , so that if is Cohen-Macaulay, then Mond's conjecture holds for ; furthermore, if is weighted homogeneous, Mond's conjecture for is
equivalent to the fact that is Cohen-Macaulay. Finally, we observe that to prove Mond's conjecture, it is enough to
prove it in a suitable family of examples.Bolsa Pesquisador Visitante Especial (PVE) - Ciˆencias sem Fronteiras/CNPq Project number: 401947/2013-0
DGICYT Grant MTM2015–64013–P
CNPq Project number 401947/2013-
A jacobian module for disentanglements and applications to Mond's conjecture
Let be a germ whose image is given by . We define an -module with the property that -, with equality if
is weighted homogeneous.
We also define a relative version for unfoldings , in such a way that specialises to when specialises to . The main result is that if are
nice dimensions, then , with equality if and only if is Cohen-Macaulay, for some stable unfolding . Here, denotes the image
Milnor number of , so that if is Cohen-Macaulay, then Mond's conjecture holds for ; furthermore, if is weighted homogeneous, Mond's conjecture for is
equivalent to the fact that is Cohen-Macaulay. Finally, we observe that to prove Mond's conjecture, it is enough to
prove it in a suitable family of examples.Bolsa Pesquisador Visitante Especial (PVE) - Ciˆencias sem Fronteiras/CNPq Project number: 401947/2013-0
DGICYT Grant MTM2015–64013–P
CNPq Project number 401947/2013-