All Questions
21 questions
2
votes
0
answers
170
views
Singular Del Pezzo of degree 2
Throughout, singular Del Pezzo means a surface with only isolated singularities and ample anti-canonical divisor.
Suppose $X$ is a singular Del Pezzo of degree 2 over a field $k$ where $\text{char}(k)\...
- 481
1
vote
1
answer
241
views
Surfaces with rational double points
Let $S\rightarrow \mathbb{P}^1$ a surface fibered in conics over a field. Assume that $S$ has a single non reduced fiber $F$ with two points of type $A_1$ on it.
Blowing-up the two points and ...
- 8,998
3
votes
1
answer
175
views
Singularities of surfaces fibered in rational curves
Let $S$ be a projective surface with a morphism $S\rightarrow\mathbb{P}^1$ whose fibers are either smooth $\mathbb{P}^1$'s or the union of two smooth $\mathbb{P}^1$'s intersecting in a point.
...
- 8,998
3
votes
1
answer
144
views
du Val singularities in Magma
Is there any way to decide whether a singularity of a surface embedded in $\mathbb{P}^5(\mathbb{Q})$ is a du Val/rational double point in Magma?
Any help is much appreciated.
- 31
4
votes
2
answers
204
views
Newton polygon notation for algebraic surface singularities
In various sources (e.g. here, Theorem 1.1 and here, Theorem 2.1 (3)), a certain notation which uses a fraction followed by a tuple is used to describe surface singularities. For example, the first ...
- 49
8
votes
1
answer
493
views
General conditions for normality of blow-up
Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along ...
- 2,032
1
vote
1
answer
93
views
Existence of meromorphic 2-forms over normal surface singularities
Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...
- 2,032
3
votes
2
answers
297
views
Singularities of a central fibre of a flat family of smooth surfaces
Suppose I have a one parameter flat family of complex surfaces (regular, of general type) whose general fibre is smooth. Is it possible for the central fibre to have singularities which are not ...
- 63
2
votes
0
answers
180
views
Pushforward of structure sheaf on quotient surface singularity
Let $f:\mathbb{C}^2 \to \mathbb{C}^2/G$ be a quotient surface singularity. What properties should $G$ have such that the pushforward $f_*\mathcal{O}_{\mathbb{C}^2}$, of the structure sheaf $\mathcal{O}...
- 1,593
5
votes
1
answer
1k
views
Are rational surface singularities $\mathbb{Q}$-Gorenstein?
I know that, in general, rational singularities are not necessarily $\mathbb{Q}$-Gorenstein. So I ask:
is there any positive result in this direction known for surfaces?
- 159
4
votes
1
answer
433
views
Infinitesimal deformations of a singular projective surface
Let $X$ be a normal projective surface with just two singular points $x_1,x_2\in X$, where $X$ has rational quotient singularities.
Assume that both the singularities in $x_1$ and in $x_2$ admit a ...
user68440
3
votes
0
answers
269
views
Hypersurface with singularities
I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?).
It was said ...
- 5,063
5
votes
1
answer
409
views
Castelnuovo's rationality criterion on singular surfaces?
Let $S$ be a projective surface over an algebraically closed field. Suppose that $q(S)=h^1(\mathcal O_S)=0$ and $P_2(S)=h^0(\mathcal O_S(2K_S))=0$. If $S$ is smooth, Castelnuovo's rationality ...
- 2,215
1
vote
1
answer
341
views
Intrinsically proving a singularity is rational
In general, how to prove a variety has rational singularities intrinsically? i.e., don't use the Artin's criterion concerning the exceptional locus. And what kinds of varieties have only rational ...
- 11
8
votes
1
answer
754
views
Why can you deform singularities in two dimensions but not in higher dimensions?
I've been trying to read this paper to understand deformations of surface quotient singularities. I'm particularly interested in when one can deform certain cyclic quotient singularities into other ...
- 3,290
4
votes
1
answer
983
views
Do there exist double points on an algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?
The title explains it all.
I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are ...
- 479
2
votes
2
answers
1k
views
singularities of the dual variety of a surface
I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...
- 3,779
3
votes
1
answer
236
views
Counting nodal singularities on a surface
How many lines in $\mathbf{P}^5$ passing through a fixed point $p$ meet in at least two points a fixed smooth surface $S$ given by the intersection of three quadrics?
Or equivalently, calling $T$ the ...
- 453
6
votes
2
answers
2k
views
Q-factorial and rational singularities on surfaces
Let $X$ be a normal surface. Is any rational singularity $\mathbf{Q}$-factorial? I've seen this somewhere for surfaces over fields, but what about the general case of an integral 2-dimensional ...
- 1,213
5
votes
1
answer
710
views
Log resolutions on surfaces and 3-folds in characteristic p
If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...
- 2,215
2
votes
1
answer
470
views
Resolution of "nice" and zero-dimensional singularities on a surface
Assume I have a singular algebraic surface $X$ over an algebraically closed field (characteristic zero if you want) which is singular in a finite set of points. I am looking for a condition as to the ...
- 4,902