All Questions
17 questions
16
votes
2
answers
4k
views
"Arithmetic genus" of a plane curve singularity.
I believe that the following questions are very basic, but I don't know how to get a reference.
Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is
...
- 14.3k
13
votes
1
answer
766
views
J.-P. Serre: Duality of regular differentials on singular curves
I already asked this on math.stackexchange.com, but didn't get any responses. I hope it is appropriate here.
Let $X'$ be an irreducible singular algebraic curve over an algebraically closed field $k$, ...
- 1,286
5
votes
1
answer
273
views
Singularities of curves that are moving
Let $k$ be an algebraically closed field, let $d\ge 2$ be an integer and let $f,g\in k[x,y,z]$ be two homogeneous polynomials of degree $d$ without common factor.
We want to know what are the ...
- 7,722
5
votes
2
answers
516
views
Tame ramification of (mild) curve singularities.
Suppose that $C$ and $D$ are curves of finite type over an algebraically closed field $k$ (we make some of these hypotheses for simplicity). We view these as pointed curves with singularities $c \in ...
- 20.5k
4
votes
1
answer
1k
views
the blowing up of a plane curve playing me tricks.
Sorry for the easy question but this is driving me crazy. Consider the blowing up of the curve $(y^2-x^3)^2+y^5$ at the origin.
On the first blowing up, on the chart that intersects the exceptional ...
- 943
4
votes
0
answers
172
views
Tangent cones to Severi strata
Let $\mathbb{C}[[x,y]]/f(x,y)$ be a reduced plane curve singularity. The base of a versal family can be taken to be (an open subset in) $\Lambda = \mathbb{C}[x,y]/(f,\partial_x f, \partial_y f)$; the ...
- 8,723
3
votes
1
answer
327
views
Segre embedding and intersections by hyperplanes
Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H_1, H_2, H_3)$ with $H_i \in H^0(\...
- 2,323
3
votes
1
answer
419
views
Is there an upper bound and a lower bound on the contribution to the genus, for a singularity of codimension k?
To make my question precise, suppose you have a complex curve locally given by
$$f(x,y) =0 $$
and $f$ has singularity of type $\chi_k$ at the origin. The codimension of
this singularity is $k$. Let ...
- 3,245
2
votes
1
answer
110
views
Determining the desingularization from the complete local ring
Suppose I have a curve $C$ over a field $k$ and that $p$ is a singular point of $C$. Let $f : X \to C$ be the desingularization of $C$ at $p$. Then for each $s \in f^{-1}(p)$ we have a map of local ...
- 2,224
2
votes
1
answer
346
views
Does the Newton polytope characterize the equisingular i.e topological type?
Whenever, people talk about singular plane curves they talk about their Newton polytope which is obviously coordinate dependent. I understand that with some conditions over the singular curve, some ...
- 115
2
votes
1
answer
259
views
Scheme-theoretic image and delta-invariants
Let $(X,o)$ be an affine, isolated, normal, Gorenstein singularity. Let $f$ and $g$ be two morphisms from $\mbox{Spec}(\mathbb{C}[[t]])$ to $X$ (also known as formal arcs) such that the closed point ...
- 1,593
2
votes
0
answers
158
views
Normalization of affine curves in singular surfaces
Let $X$ be a normal, isolated surface singularity with $x_0 \in X$ the unique singularity. Let $C \subset X$ be a hyperplane section i.e., defined by a single equation. Denote by $n:\widetilde{C} \to ...
- 2,032
2
votes
0
answers
186
views
Singularities of algebraic curves, and torsion of the pull-back of the differential module by the normalisation
The problem in the following :
given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field $\...
- 223
1
vote
1
answer
223
views
Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?
Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
- 2,126
1
vote
1
answer
174
views
Is the space of degree $d$ curves with marked smooth points dense inside the space of curves with marked points?
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d} $ be the space of nonzero
homogeneous degree $d$ polynomials in three variables upto scaling, where
$\delta_d = \frac{d(d+3)}{2} $
(basically degree $...
- 3,245
1
vote
0
answers
178
views
What are algebroid curves/branches and their value semigroup?
In “The moduli problem for plane branches”, by O. Zariski, the author defines a plane branch as an irreducible element $f \in \mathbb C[[x,y]]$. In the more recent article "The semigroup of a ...
- 111
1
vote
0
answers
146
views
Is there any explicit result on the triangulated category of singularities of a curve?
This question is related to this MO question.
Let $X$ be a projective curve over a field $\mathbb{C}$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category ...
- 9,019