Questions tagged [algebraic-surfaces]
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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Kodaira classification and the McKay correspondence
Kodaira's table of singular fibers has a singular fiber
for each of $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$; these are chains or cycles of (-2)-curves connected ...
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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 ...
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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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Bezout Theorem in $\mathbb P^3$
For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ curve passes then $d^2\geq m_1^2+m_2^2+\...
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Uniformization of Kodaira fibered surfaces
Consider a Kodaira fibration. i.e. a smooth non-isotrivial fibration $X\rightarrow C$ with $X$ a smooth complex surface and $C$ a smooth complex curve, such that both the genus of $C$ and genus of the ...
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When does Bogomolov's inequality become an equality?
The Bogomolov theorem says if $V$ is a rank 2 vector bundle on an algebraic surfaces $S$ is $H$-stable (in the sense of Mumford-Takemoto) for some ample divisor $H$, then $c_1^2(V) \leq 4c_2(V)$ holds....
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A question on an elliptic fibration of the Enriques surface
Let $S$ be an Enriques surface over complex numbers. It is known that $S$ admits an elliptic fibration over $\mathbb{P}^1$ with $12$ nodal singular fibers and $2$ double fibers. How can I see this ...
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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...
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What does Hodge theory tell us about simply connected surfaces of general type
Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...
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A question on existence of degeneration of Enriques surface.
Let $S$ be an Enriques surface, i.e. a quotient of a K3 surface by a free involution. Enriques surfaces arise as elliptic fibrations $S\rightarrow \mathbb{P}^1$ with 12 singular fibers and 2 double ...
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The central fiber of this family of surfaces?
I have a question on a description of a central fiber of the following family of surfaces.
Let $B,C \subset \mathbb{P}^2$ be smooth curve of degree $2d$ and $d$ respectively. Let's consider the ...
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What can a quartic surface in $\mathbb{P}^3$ with an ordinary quadruple point look like?
All varieties will be projective and over $\mathbb{C}$.
If $S$ is any surface in $\mathbb{P}^3$ of degree 2 that posseses an ordinary double point, it follows easily that $S$ is projectively ...
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Surfaces ruled over elliptic curves
Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve.
Suppose $E$ is an elliptic curve ...
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Basics of minimal Elliptic Surfaces [following Beauville]
I am reading Beauville's chapter IX on Elliptic surfaces.
Let $S$ be a minimal elliptic surface with $\kappa=1$ and $p:S\rightarrow C$ be the elliptic fibration.
We know $K^2=0$. Suppose the $m$-...
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Embeddings of smooth projective surfaces
Let $X$ be a smooth projective surface not contained in $\mathbb{P}^3$.
Is there any known condition on $X$ under which I can embed it into $\mathbb{P}^3$ such that the its image contains at most ...
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pencils on varieties of general type
I was wondering about a generalization of the following property of surfaces of general type.
Let $X$ be a smooth projective surface of general type. Then there is no pencil of rational or elliptic ...
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Existence of smooth surfaces containing a curve
Let $C$ be a curve in $\mathbb{P}^3$, possibly non-reduced. Assume, there exists a smooth surface in $\mathbb{P}^3$ containing $C$. Is it true that for $d \gg 0$, a generic element of $I_d(C)$ defines ...
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families of curves on surfaces which are products of curves
Let $C$ be a projective curve (over an algebraically closed field) of genus $\geq 1$. Let $S = C \times C$. By normalisation we have a ramified cover $C \to \mathbb{P}^1$ and so a map $p: S \to \...
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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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Is the set of surfaces over Spec Z with ample canonical sheaf empty
Main question. Does there exist a smooth projective morphism $X\to$ Spec $\mathbf Z$ of relative dimension two such that the canonical sheaf $\omega_{X_{\mathbf Q}}$ of the generic fibre $X_{\mathbf Q}...
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Quotients of rational surfaces
Let $X$ be a projective surface defined over a field $k$ of characteristic $0$, and let $G$ be a finite group acting biregularly on $X$.
Assuming that $X$ is rational over $k$, is the quotient $X/G$ ...
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Contracting rational curves on surfaces and getting something non-algebraic
Recently I posted an "announcement" on arxiv where I said something to the effect of "this is the first example we know where contracting (a tree of) rational curves from a non-singular algebraic ...
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Vector Bundles on normal surfaces
Let $X$ be a projective normal surface over $\mathbb{C}$. In this related question it is stated as soon as $X$ is smooth any vector bundle defined on the compliment of a codimension 2 subset extends ...
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Hilbert scheme of 2 points on an elliptic curve
The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$.
What can we say about the ...
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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 ...
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Constructing a curve with good reduction over a function field
Let $K$ be the function field of a smooth projective connected curve $B$ over $\mathbf{C}$.
Let $g\geq 0$ be an integer.
Does there exist an nonsingular integral $\mathbf{C}$-scheme $X$ with a ...
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Bertini's Theorem small print
Suppose $S\subset \mathbb{P}^n$ is a smooth del Pezzo surface and $C$ is an irreducible smooth curve (you can make it rational if it simplifies the setting) such that $\mathcal{L}=\vert -K_S-C\vert $ ...
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How to construct Enriques surface from Fermat K3
Let $x_1^4+x_2^4+x_3^4+x_4^4=0 \subset \mathbb{P}^4$ be the Fermat K3 surface. Is it possible to start from some involution on $\mathbb{P}^3$, do blow-ups to get rid of fixed points and then quotient ...
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Invariant lattice of algebraic surface.
Given an algebraic surface $S$ with action of a finite group $G$. Is it true that the invariant lattice $H^2(X,\mathbb{Z})^G$ is generated by elements pulled back from the $H^2(X/G,\mathbb{Z})$ (or $H^...
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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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Genus two pencil in K3 surface
It is known that smooth $K3$ surface can be obtained as two fold branched cover of rational elliptic surface $E(1) = \mathbb{CP}^2 9 \bar{{\mathbb{{CP}^2}}}$ along the smooth divisor $2F_{E(1)} = 6H - ...
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Question on K3 Surface
Is it possible to realize $K3$ surface as a ramified double cover of rational elliptic surface? If so, is there way to see an elliptic fibration structure on $K3$ from such cover? It seems to me one ...
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Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
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The cohomology of the relative dualizing sheaf of a relative curve
Let $X\to S$ be a curve over $S=\mathrm{Spec} \ \mathbf{Z}$. Let $\omega$ be the relative dualizing sheaf of $X\to S$. Let $g$ be the genus of the generic fibre. Assume that $g\geq 2$.
I know that $\...
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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 ...
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