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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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8 votes
1 answer
796 views

What is the smallest and "best" 27 lines configuration? And what is its symmetry group?

I was this past year working with a bright high-schooler on algebraic geometry following Reid's book Undergraduate Algebraic Geometry, and we got all the way to proving that there is at least one line ...
1 vote
2 answers
196 views

What are the finite étale coverings of a quasi-hyperelliptic surface?

Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial. Question: Is there a finite étale covering $Y \rightarrow X$ such that $Y$ is an abelian ...
2 votes
1 answer
268 views

Jacobian fibration of elliptic fibration: basic relations between Enriques invariants

Let $f: X \to B$ be an elliptic fibration, so proper map from smooth surface $X$ onto smooth conn. curve over alg closed base field $k$ with connected fibers such that almost all fibers are elliptic ...
0 votes
0 answers
101 views

Image of K3 surface under finite map with pure ramification rational

Let $X$ be a projective K3 surface and $f: X \to Y$ a non etale, finite map, restricting to etale on non empty open $U \subset Y$ of degree prime to char of alg closed base field of $k$. Assume ...
4 votes
1 answer
175 views

Every elliptic surface contains only finitely many negative self-intersection rational curves?

By a properly elliptic surface, I mean an algebraic surface $X$ with Kodaira dimension $\kappa(X)=1$. It has a natural elliptic fibration $\pi\colon X\rightarrow S$. According to section 5.2 of this ...
1 vote
0 answers
117 views

Quotient of K3 surface: complex vs positive characteristic

Let $f: X \to X$ be a non-symplectic automorphism of finite order of complex projective K3 surface $X$. (Recall: Non-symplectic means that the induced action on $H(X,K_X)=H^0(X, \Omega_X^2)$ is not ...
0 votes
0 answers
112 views

Irregularity of surfaces for dominant maps

I have a question about an argument in the proof of Lemma 1.2.(1) in Quotients of K3 surfaces modulo involutions by D. Q. Zhang: Let Let $(X, \sigma)$ be X be a smooth projective K3 surface with an ...
0 votes
1 answer
128 views

Chern Classes of $\mathcal{O}_E(1)$ on $\mathbb{P}(E)$ for $E = \mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$

Let $E =\mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$ and denote by $\mathcal{O}_E(1)$ the dual of the tautological bundle. How can I compute $c_1^2(\mathcal{O}_E(1)), c_1^3(\mathcal{O}_E(1))$, $...
1 vote
0 answers
219 views

Quotient of K3 surfaces by non-symplectic automorphism of finite order

Let $X$ be a $K3$ surface and $f: X \to X$ a non-symplectic morphism (ie non symplectic in sense of that that the induced action on $H(X,K_X=H^0(X, \Omega_X^2)$ is not trivial) of finite order. ...
0 votes
0 answers
99 views

Quotients of K3 surfaces vs cyclic covers

Let $X$ be an algebraic K3 surface (for sake of simplicity, with base field of char $\neq 2$) and $f: X \to X$ a non-symplectic morphism (i.e. non-symplectic in sense of that that the induced action ...
0 votes
0 answers
110 views

Lefschetz Theorem in Dolgachev's On automorphisms of Enriques Surfaces

Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces. This 2.1. Proposition. states ...
0 votes
0 answers
89 views

Contraction of extremal ray on a smooth projective threefold

I have some issues about understanding the contraction of extremal ray in a concrete situation: Let $\mathcal{E}=\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1\times \...
0 votes
0 answers
127 views

Relative minimal models of pencils of surfaces

I would like to ask for recomendation for literature on theory relative minimal models of surfaces, where "relative" in sense of that the study objects are not surfaces alone (="absolue ...
1 vote
0 answers
161 views

Special elliptic pencil of an Enriques surface (arguments in a proof)

I have a couple of questions about arguments in the proof of Lemma 2.6 (see absol page 199, rel p 9) from Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups: The setup: Let $Y$ ...
1 vote
0 answers
139 views

Estimation of the degree of a projective surface

Let $S \subset \mathbb{P}^{n}_{\mathbb{C}}$ a projective integral smooth surface lying in no hyperplane (I adopt the point of view of scheme). I will denote by $d$ its degree. The following questions ...
3 votes
1 answer
163 views

Is a pseudo-effective divisor on a rational surface numerically effective?

Let $D$ be a pseudo-effective $\mathbb{R}$-divisor on a rational surface. Can we find an example that the numerical class of $D$ contains no effective divisor?
3 votes
1 answer
211 views

Existence of a universal $0$-cycle

For a smooth (complex) projective surface $S$ with non trivial Albanese variety, are there simple conditions for the existence a universal codimension $2$ cycle i.e. a $Z\in \mathrm{CH}^2(Alb(S)\times ...
0 votes
1 answer
311 views

Triple covers of $\mathbb{P}^2$ with Tschirnhausen module $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$

Let $X$ be a surface as in the title. Rick Miranda said that $X$ is a Steiner cubic in $\mathbb{P}^4$, and the cover map is projection. Invariants of $X$ can be computed directly, $p_g(X)=0,K^2_X=8,e(...
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 ...
5 votes
0 answers
145 views

Symmetric groups acting on rational surfaces

Let $X$ be a complex projective rational surface. Is there an upper bound on $n\in\mathbb{N}$ such that $S_n\subset \text{Aut}(X)$? Here $S_n$ is the symmetric group on $n$ elements.
0 votes
0 answers
51 views

Does this tangent developable-like surface have a cusp along a curve or is it smooth?

Consider the following surface $X$ which is a subvariety of the full flag variety $Y=\{0 \subset V_1 \subset V_2 \subset V\}$ where $V$ is a fixed three-dimensional vector space and ${\rm dim} V_i =i$....
3 votes
1 answer
271 views

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 $...
2 votes
0 answers
149 views

Non-proper intersection between divisors on $\mathbb{P}^1$-bundle of Hirzebruch surfaces

We are working on algebraic closed field $k$. Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$, $C_0$ and $C_{\infty}$ are its zero and infinity sections ...
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,\...
6 votes
2 answers
276 views

Embedding degree 1 Del Pezzo surfaces in $\mathbb{P}(1,1,2,3)$

In the projective bundle $\mathbb{P}(\mathcal{O}(-1)\oplus \mathcal{O}(-1)\oplus \mathcal{O})\rightarrow\mathbb{P}^1$ consider the hypersruface $$ X := \{a_{00}y_0^2+a_{01}y_0y_1+a_{02}y_0y_2+a_{11}...
2 votes
1 answer
137 views

Contraction of $(-1)$ curve and extremal ray

I want to prove Castelnuovo's contraction theorem by Mori's contraction theorem. Question. How can one show that a $(-1)$ curve on a smooth surface is an extremal ray?
5 votes
1 answer
396 views

Articles of Casnati on algebraic varieties

I am attempting to track down online copies of the following two algebraic geometry articles. Is there some repository where these might be found? If necessary I could use the first few pages of each ...
1 vote
0 answers
47 views

Positivity of self-intersection of dicisor associated to meromorphic function

In the book "Holomorphic Vector Bundles over Compact Complex Surfaces" by Vasile Brînzănescu, in the proof of theorem 2.13 there is the following claim Let $X$ be a compact non-algebraic ...
2 votes
0 answers
95 views

Reference request The support of $f$-nef divisor

I'm seaching for a proof of the theorem below. Do you know any reference? Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...
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 $\...
14 votes
2 answers
3k views

Motivation for birational geometry

I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
1 vote
0 answers
109 views

One question about Manetti surface

I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof. Theorem 5.2 states that fixed a ...
3 votes
2 answers
429 views

Question about surface singularities

Throughout, $X$ will be a projective surface. I am looking for examples of the following surface singularities, I) A rational singularity that is not quotient. Obviously, it has to be non-Gorenstein, ...
8 votes
3 answers
3k views

Cone of curves and Mori theorem for algebraic surfaces

In describing part of the geometry of the cone of curves for an algebraic surface $S$, we need to find $(-1)$ curves within $S$. Once we've done that, then we can say that the "negative" ...
2 votes
0 answers
120 views

Equality case of the log-Bogomolov-Miyaoka-Yau inequality

The Bogomolov-Miyaoka-Yau inequality for sufaces says that if $X$ is a smooth projective minimal surface of general type then $c_1(X)^2 \le 3 c_2(X)$. It is a theorem of Yau (I think) that equality ...
1 vote
0 answers
200 views

Action on Enriques surface by sections of Jacobian fibration

A question about a statement in Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups: The setup: Let $\pi: Y \to \mathbb{P}^1$ be a special elliptic pencil of complex Enriques ...
3 votes
2 answers
296 views

Multiplicity of irreducible component of a singular fiber of a $\mathbb{P}^1$-fibration

Let $X$ be a smooth projective surface and $f:X\to\mathbb{P}^1$ be a $\mathbb{P}^1$-fibration with a singular fiber consisting of a tree with three irreducible rational ($-2$)-curves $D_1$, $D_2$, $...
1 vote
0 answers
212 views

Formula for Pushforward of structure sheaf for branched coverings

I have some questions of same flavour about two following constructions in Daniel Huybrechts's notes on K3 surfaces. Construction 1: Kummer surface (Example 1.3 (iii), page 8) Let $k$ be a field of $...
1 vote
0 answers
181 views

On the positivity of cotangent bundle of elliptic surfaces

I am working on the complex numbers field $\mathbb{C}$, for simplicity. However you can relax this assumption if you want. Let $X$ be an elliptic surface, id est there is a proper morphism $\pi\colon ...
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 ...
2 votes
1 answer
732 views

Where to find "Families of curves on a surface of general type" (MR0457450)?

I am currently doing some research on surfaces of general type and I need some results from Bogomolov's paper: Bogomolov, F. A. Families of curves on a surface of general type. Dokl. Akad. Nauk SSSR ...
2 votes
1 answer
234 views

(Non-)Rationality of a certain quotient of the symmetric square of the Fermat sextic (quartic) curve

Consider the Fermat sextic curve $F: x^6 + y^6 + 1 = 0$ over an algebraically closed field of characteristic $0$. It has the two order $3$ automorphisms $\omega_x(x,y) := (\omega x, y)$ and $\omega_y(...
1 vote
1 answer
235 views

Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface

Let $k$ be an algebraically closed field of $\text{ch}(k) =0$. Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack. Let $\mathbf{P}(1,1,2)$ be the weighted ...
3 votes
1 answer
343 views

Existence of elliptic curves on surfaces of general type

Let $X$ be a complex minimal surface of general type, id est $K_X$ is big and nef. It is well-known that $\displaystyle\int_X3c_2(X)-c_1(X)^2\geq0$, and the equality holds if and only if $X$ is ...
1 vote
0 answers
97 views

On surfaces of general type wich saturate the BMY-inequality

Let $\mathbb{K}$ an algebraically closed field of characteristic $0$, let $X$ be a smooth minimal surface of general type. It is known that surfaces satisfy, among other thing, the (Bogomolov-Miayoka-...
3 votes
1 answer
888 views

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 ...
1 vote
0 answers
98 views

Does there exist a simply connected surface with CM whose cotangent bundle is ample?

Does there exist a smooth projective complex surface $X$ such that, (1) $\pi_1(X) = 0$ (2) $\Omega_X^1$ is ample (3) the Mumford-Tate group of $H^2(X)$ is a torus There exist examples with any two of ...
8 votes
1 answer
255 views

Can "fake rational surfaces" be simply-connected?

I say that a smooth projective complex algebraic surface $X$ is a "fake rational surface" if its Hodge diamond looks like: and $X$ is of general type. It is well-known that fake projective ...
4 votes
0 answers
100 views

Fundamental groups of Hirzebruch's line arrangement varities

Let $\Lambda$ be a line arrangement in $\mathbb{P}^2$ and $n > 0$ an integer. Then Hirzebruch defined a smooth projective surface $H(\Lambda, n)$ as the minimal desingularization of a covering $Y \...
1 vote
0 answers
73 views

Explict equations for unirational Enriques surface with a nonzero 1-form

I am hoping to write down very explicitly the equations for the following data: an Enriques surface $X$ of type $\mathrm{Pic}^{\tau} = \mathbb{Z} / 2 \mathbb{Z}$ such that its canonical $\mu_2$-cover ...

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