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Results tagged with algebraic-surfaces
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user 4096
An algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
3
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1
answer
262
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degeneration of a Veronese surface
Let $V$ be the Veronese surface, obtained as the image of $\mathbb{P}^2$ in $\mathbb{P}^5$ by the complete linear system of conics. I understand that $V$ can degenerate to the union of a cubic scroll …
- 39.3k
2
votes
2
answers
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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 dua …
- 1,803
16
votes
1
answer
4k
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what is the cyclic cover trick?
What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explanation, both talking about curves and surfaces...
- 22.3k
3
votes
2
answers
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Quartic symmetroids and 10-points sets
A quartic surface in $\mathbb{P}^3$ is said to be a "symmetroid" if its equation is obtained as the determinant of a 4x4 symmetric matrix of linear forms. It is well known that the general symmetroid …
- 66.3k
6
votes
1
answer
370
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Self intersection and deformations
Suppose I have a smooth projective surface $S$ and a curve $C\subset S$. The self-intersection of $C$ is by definition the degree of the restriction to $C$ of the normal bundle of $C$ inside $S$.
By …
- 66.3k
0
votes
0
answers
115
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invariance of the dimension of severi varieties of surfaces
Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves on $S$ belonging to $|L|$, …
- 3,779
3
votes
singularities of the dual variety of a surface
The answer is yes, the claim is true. Moreover, as long as the dual variety is an irreducible hypersurface (i.e. most cases), then the claim on the singularity of the plane section holds true for any …
- 3,779
3
votes
1
answer
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octic K3s inside cubic 4-folds
From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could …
- 1,334
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0
answers
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quasi-trigonal curves
I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory …
- 3,779