# Tagged Questions

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.

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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 $|L|$, ...
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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 ...
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### Dual of a Complex 2-Torus

Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?
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### Reference for Automorphisms of K3 surfaces

I am looking for some introductory reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion?
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### Complete Linear system on Del Pezzo surfaces

Is there always a reducible curve (EDIT: with exactly two irreducible components intersecting in at least 2 points) in a complete linear system (EDIT: of dimension at least 2 with curves of genus at ...
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### On complex surfaces with Kodaira dimension 1

Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1$. What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...
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### Contracting a curve of negative self-intersection on a surface

It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermore, I know of ...
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### A classification of rational surfaces with effective $K$

I would like to know if there can be some kind of classification of normal rational surfaces with Gorenstein singularities, such that their canonical divisor is effective. Additional question. Are ...
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### Divisor class group on blowup of nodal surface

The following got no answer on mathstackexchange. I believe it not to be hard, but maybe it is a little specialized? All varieties will be over $\mathbb{C}$ and projective unless stated otherwise. ...
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### 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 ...
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### Absorbing ramification and factoring finite flat maps

In his Algebraic surfaces book, Beauville gives a result allowing one to "absorb ramification" for certain maps (see below). There are also something similar one can do with number fields. I would ...
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### Can you get an Enrique surface from quotient of Abelian surface?

Let $A=\mathbb{C}^2/\Lambda^2$, where $\Lambda=\mathbb{Z}+i\mathbb{Z}$, be an abelian surface. Then every body knows that the resolution of the quotient $A/<\pm>$ is a K3 surface. Question: Is ...
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### blowing up general k points on the plane

Del Pezzo surfaces are obtained by blowing up $1 \leq k \leq 8$ points on general position in $\mathbb{P}^2$. What does it happen when the number of points is larger than nine? In this sense, ...
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### Divisorial contraction: when is the image an algebraic space or a stack?

Let $X$ be a smooth projective surface (in the category of varieties, or schemes), and let $C\subset X$ be a curve (a priori not irreducible, but the irreducible case in itself is already interesting)....
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### Question on Ball Quotients

Let $X$ be a compact Kahler surface which is a ball quotient. Can such $X$ contain a torus $T$ such that the fudamental class of $T$ is non-trivial? I expect this is false as $\pi_{1}(X)$ is a ...
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### Algebraic surfaces of general type

Question. Are there smooth complex surfaces of general type with an irregularity $q = 1$ and Euler characteristic $3$. If the answer is yes, what is known about the geometry of such surfaces? Are ...
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### On $\pi_1$ of surface of general type

Let $X$ be an algebraic surface of general type. Assume $K_X$ is an integer multiple of another class $A$, and the class $A$ can be represented by a symplectic submanifold $S$ of $X$ with non-negative ...
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### An arithmetic analogue of the discriminant curve of a conic bundle threefold

I am looking for an "arithmetic" analogue of a well known result on threefolds with a conic bundle structure. The following result can be found in [Iskovskikh - On the rationality problem for conic ...
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### 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 ...
### Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$
Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d? e.g. for $d=4$ the cohomology ...
Let $k$ be an algebraically closed field of characteristic $p>0$. I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...