All Questions
Tagged with algebraic-surfaces at.algebraic-topology
12 questions
4
votes
1
answer
296
views
Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset
Let $X$ be a normal algebraic surface (over $\mathbb{C}$) and $Y$ its smooth locus, i.e., the complement of the singularities of $X$. Suppose $Z\subset Y$ is a Zariski open subset of $X$. Then is it ...
- 335
4
votes
1
answer
218
views
Topological interpretation of the canonical cover of a logarithmic Enriques surface
A normal projective surface $Z$ with at worst quotient singularities is called a logarithmic (log) Enriques surface if its canonical Weil divisor $K_Z$ is numerically equivalent to zero, and $H^1(Z,\...
- 213
2
votes
0
answers
55
views
Fundamental group of cyclic branched cover of affine plane
Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
- 615
9
votes
1
answer
284
views
Fundamental group of a smoothing of a complex surface
Let $X_0$ be a compact complex algebraic surface with an isolated singularity and let $X_t$ be a smoothing of $X_0$ over the disc. How can we compute the fundamental group of $X_t$ say in terms of the ...
- 3,290
1
vote
0
answers
78
views
Diffeomorphism induced by small perturbation
Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}.
\end{...
- 521
6
votes
1
answer
487
views
Topology change induced by small perturbation
Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:f_{\epsilon}(\vec x)\equiv\epsilon ((x^2 + y^2 - ...
- 521
7
votes
1
answer
2k
views
Relating the holomorphic Euler characteristic of a family of algebraic varieties to properties of the base and fibers
Let $f : X\rightarrow Y$ be a proper flat morphism (of schemes) with connected fibers over a smooth projective curve $Y$ over $\mathbb{C}$. Let $X_{y_0}$ denote a smooth fiber over $y_0\in Y$.
If $f$ ...
- 10.7k
4
votes
1
answer
639
views
Intersection form in Algebraic Geometry/Topology
Let $S$ be a smooth complex projective surface. We let define an intersection form $(-)\cdot(-)$ on $\mathsf{Pic}(S)$ by setting $$D\cdot D':=\mathcal{O}_S(D)\cdot\mathcal{O}_S(D')$$ where the ...
- 1,252
5
votes
1
answer
461
views
Homeomorphism between del Pezzo surfaces
Let $X$ and $Y$ be smooth del Pezzo surfaces of the same degree $K_X^2=K_Y^2$.
Are the sets $X(\mathbb{C})$ and $Y(\mathbb{C})$ homeomorphic, or at least homotopy equivalent?
- 53
23
votes
1
answer
718
views
Del Pezzo surfaces and homotopy groups of spheres
A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...
- 6,920
8
votes
1
answer
652
views
How do branched coverings of complex surfaces "fit" with branched coverings of curves?
Since I'm used to working with algebraic $\pi_1$'s, which don't work well with surfaces, I find myself lacking geometric intuition when I attempt to do these types of purely geometric arguments. I'm ...
- 5,929
6
votes
4
answers
873
views
Interaction of topology and the Picard group of Algebraic surfaces
It is well known that a smooth cubic surface $X\subset \mathbb{P}^3$ has exactly 27 lines in it. Furthermore, it is easy to check that Picard group $$Pic(X)\cong \mathbb{Z}^7$$ Here the generators are ...
- 2,132