Singularity Theory and Algebraic Geometry
http://hdl.handle.net/20.500.11824/18
2023-10-21T17:02:50ZMonodromy conjecture for semi-quasihomogeneous hypersurfaces
http://hdl.handle.net/20.500.11824/1620
Monodromy conjecture for semi-quasihomogeneous hypersurfaces
Blanco, G.; Budur, N.; Van der Veer, R.
We give a proof the monodromy conjecture relating the poles of motivic zeta
functions with roots of b-functions for isolated quasihomogeneous hypersurfaces, and more
generally for semi-quasihomogeneous hypersurfaces. We also give a strange generalization
allowing a twist by certain differential forms.
2021-01-01T00:00:00ZPolar exploration of complex surface germs
http://hdl.handle.net/20.500.11824/1613
Polar exploration of complex surface germs
da Silva, A.B.; Fantini, L.; Némethi, A.; Pichon, A.
We prove that the topological type of a normal surface singularity
pX, 0q provides finite bounds for the multiplicity and polar multiplicity of pX, 0q,
as well as for the combinatorics of the families of generic hyperplane sections
and of polar curves of the generic plane projections of pX, 0q. A key ingredient
in our proof is a topological bound of the growth of the Mather discrepancies
of pX, 0q, which allows us to bound the number of point blowups necessary to
achieve factorization of any resolution of pX, 0q through its Nash transform.
This fits in the program of polar explorations, the quest to determine the generic
polar variety of a singular surface germ, to which the final part of the paper is
devoted.
2021-01-01T00:00:00ZThe dimension of the image of the Abel map associated with normal surface singularities
http://hdl.handle.net/20.500.11824/1605
The dimension of the image of the Abel map associated with normal surface singularities
Nagy, J.; Némethi, A.
Let (X, o) be a complex normal surface singularity with rational homology sphere link
and let Xe be one of its good resolutions. Fix an effective cycle Z supported on the exceptional
curve and also a possible Chern class l
′ ∈ H2
(X, e Z). Define Ecal
′
(Z) as the space of effective
Cartier divisors on Z and c
l
′
(Z) : Ecal
′
(Z) → Picl
′
(Z), the corresponding Abel map. In this note
we provide two algorithms, which provide the dimension of the image of the Abel map.
Usually, dim Picl
′
(Z) = pg, dim Im(c
l
′
(Z)) and codim Im(c
l
′
(Z)) are not topological, they are
in subtle relationship with cohomologies of certain line bundles. However, we provide combinatorial
formulae for them whenever the analytic structure on Xe is generic.
The codim Im(c
l
′
(Z)) is related with {h
1
(X, e L)}L∈Im(c
l
′
(Z)); in order to treat the ‘twisted’
family {h
1
(X, e L0 ⊗ L)}L∈Im(c
l
′
(Z)) we need to elaborate a generalization of the Picard group and
of the Abel map. The above algorithms are also generalized.
2019-01-01T00:00:00ZMonodromy conjecture for log generic polynomials
http://hdl.handle.net/20.500.11824/1604
Monodromy conjecture for log generic polynomials
Budur, N.; Van der Veer, R.
A log generic hypersurface in P
n with respect to a birational modification of
P
n
is by definition the image of a generic element of a high power of an ample linear series
on the modification. A log very-generic hypersurface is defined similarly but restricting
to line bundles satisfying a non-resonance condition. Fixing a log resolution of a product
f = f1 . . . fp of polynomials, we show that the monodromy conjecture, relating the motivic
zeta function with the complex monodromy, holds for the tuple (f1, . . . , fp, g) and for the
product fg, if g is log generic. We also show that the stronger version of the monodromy
conjecture, relating the motivic zeta function with the Bernstein-Sato ideal, holds for the
tuple (f1, . . . , fp, g) and for the product fg, if g is log very-generic. Even the case f = 1 is
intricate, the proof depending on nontrivial properties of Bernstein-Sato ideals, and it singles
out the class of log (very-) generic hypersurfaces as an interesting class of singularities on
its own.
2020-01-01T00:00:00Z