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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bound on the genus of a fiber of the Albanese map of a surface with $h^1({\mathcal O})=1$?

This is maybe more an open problem than a question, since I have seriously thought about it and asked several people working on algebraic surfaces with no success. I hope somebody here can ...
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A question on surfaces in $\mathbb{P}^4$

On surfaces in $\mathbb P^4$,Ellingsrud and Peskine has proved that There exists an integer $d_0$ such that for any integer $d>d_0$,any smooth surface of degree $d$ in $\mathbb P^4$ is of ...
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Ample divisors on projective surfaces

Question: If $X$ is a projective surface and $U$ is an open affine subset of $X$, then is it true that $X \setminus U$ is the support of an (effective) ample divisor on $X$? Background: I was reading ...
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Curves on rational surfaces and Lang's conjecture for M_g

There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not ...
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Surfaces with $q=2$ and generically finite Albanese map

I have a family of surfaces of general type $S$ with $q(S)=2$, and such that the Albanese map $$\alpha \colon S \longrightarrow A:=\mathrm{Alb}(S)$$ is generically finite of degree $n$. By a result of ...
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11 votes
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Singular curve on an abelian surface

Let $C_2$ be a smooth genus $2$ curve and $J(C_2)$ its Jacobian. It is well known that the blow-up of $J(C_2)$ at the origin $o$ is isomorphic to the second symmetric product $\textrm{Sym}^2(C_2)$, ...
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9 votes
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How many characteristics is a random surface unirational in?

Suppose I have a surface $X$ defined over $\mathbb{Z}$. I am interested in the set $S_X$ of primes $p$ such that $X_{\overline{\mathbb{F}}_p}$ is unirational. If I choose a "random" surface ...
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8 votes
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Very ample divisors on blow ups of the projective plane

Let $X$ be $\mathbb{P}^2$ blown up at $k$ points in general position. The Picard group of $X$ is just $\mathbb{Z}^{k+1}$ and one knows the intersection product explicitly. If $D$ is an ample divisor, ...
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6 votes
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Weyl group and Galois action on cubic surfaces

Let $X$ be a smooth cubic surface over a field $k$. Denote by $\bar{k}$ the separable closure of $k$ and $\bar{X}:=X\times_{k}\bar{k}$. Then it is well know that there exists a homomorphism $$ \phi:\...
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nonvanishing higher cohomology of a very ample divisor

I am looking for smooth projective varieties $X$, with $h^i(X, \mathcal{O}_X) = 0$ for $i > 0$, with a very ample line bundle $L$ with some nonvanishing higher cohomology. What is clear: (1) Curves ...
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Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?

For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$. If ...
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How does the "todd class operator" commute with Nakajima's q operators on Hilbert schemes of points on surfaces

Let $S$ be a surface, and $S^{[n]}$ the Hilbert scheme of $n$ points on $S$. One defines $$ \mathbb H = \bigoplus_n H^*(S^{[n]}). $$ One has Nakajima's operators $\mathfrak q_n(\alpha)$ (with $\alpha \...
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Are all these K3 surfaces supersingular?

Consider all the smooth K3 surfaces given by $X^4+W^2X^2+XW^3 = f(Y,Z,W)$ or $X^4+XW^3 = g(Y,Z,W)$ over $\mathbb F_{2}$ with $f$ or $g$ homogenous of degree 4. There are a lot of choices for $f$ and $...
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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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5 votes
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Deformations of a blow up

My question is related to this question, but I'm looking for something a bit more explicit. Let $S$ be a smooth surface over $\mathbb C$, fix a point $s\in S$ and take the blow up $\beta \colon S' \...
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5 votes
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156 views

Arnold's theorem on small denominators and holomorphic tubular neighborhoods

By a theorem of Grauert, along a curve with negative self-intersection a complex surface is locally biholomorphic to a neighborhood of the zero section of that curve inside its normal bundle. For ...
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5 votes
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Map associated to linear system onto curve is morphism

In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
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5 votes
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256 views

Max Noether's theorem for algebraic surfaces

The well-known Max Noether's theorem for curves [see Arbarello-Cornalba-Griffiths-Harris Geometry of Algebraic Curves vol.1, p. 117] states that, if a smooth curve $C$ is non-hyperelliptic, then the ...
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5 votes
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How to prove a statement on weak del pezzo surface?

Let $X$ be a weak del pezzo surface($-K_X$ is nef and $K_X^2>0$) with $3\leq K_X^2\leq 5$. Let $H$ be a $1$-class on $X$ such that $H.C\geq 1$ for any $C\in I^{irr}(X)$. Then $2H+K_X$ is effective ...
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Is a Kummer surface over an finite field $\mathbb{F}_q$ supersingular iff $\mathbb{F}_q$-unirational?

Let $A$ be an abelian surface over an finite field $\mathbb{F}_q$. In particular, I am interested in the case when $A$ is a Jacobian variety. Is the Kummer surface $K_A/\mathbb{F}_q$ Shioda-...
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Adjunction map for projective surfaces

Before stating my question, let me recall (part of) the classical result on the adjunction map for complex projective surfaces, due in this modern form to Beltrametti and Sommese: Adjunction ...
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5 votes
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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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5 votes
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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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4 votes
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Reducible surface as a degeneration

I am interested in the following situation. If $S_1\cup_D S_2$ is a union of two irreducible smooth projective surfaces over $k=\overline{k}$(over $k=\mathbb{C}$ is enough, if it's relevant) glued ...
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Is there a way to calculate the Picard $\mathbb{F}_q$-number of an (rational or K3) elliptic surface?

Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface $$ \mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6, $$ where $a_i(t) \in \mathbb{F}_{q}[t]$. Is there a way ...
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4 votes
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199 views

Del Pezzo surfaces and Picard--Lefschetz theory

Let $X$ be a del Pezzo surface, say of degree $3$ for concreteness. Then compare: the $27$ $(-1)$-curves form a lattice $E_6\subseteq H^2(X;\mathbf{Z})$; the Weyl group is generated by the simple ...
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Fibered surfaces degenerating to Frobenius

Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
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4 votes
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141 views

Is there a $\sum e_if_i=n$ in higher dimensions?

If $X\to Y$ is a finite map of connected proper algebraic curves over a field, then for any point $y\in Y$, the sum $\sum e_xf_x=n$ of ramification times inertia degrees over points $x$ mapping to $y$ ...
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  • 4,004
4 votes
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163 views

Can Kummer surfaces coming from the same abelian surface be Cremona equivalent / isomorphic?

Assume we are given a simple abelian surface $A$ which has 2 non-equivalent principal polarizations $D_1$ and $D_2$ in $NS(A)$ (up to isomorphism), thus giving rise to two non-isomorphic smooth ...
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4 votes
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Finding Rational Curves on a Surface

Let the field of rational numbers be our base field $k$. I hope to find all rational curves on the following surface $S$ defined by $f$. You can find the motivation in the end. $f= (x^2y^2)z^3 + (5x^...
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4 votes
0 answers
165 views

How close is $h^0(mD)$ to be a polynomial?

Let $X$ be a normal (or smooth if it helps) projective variety over an algebraically closed field $k$. Fix a Cartier divisor $D$: I am interested in knowing how $h^0(mD)$ behaves as $m$ varies. At ...
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Are there methods to compute the induced action of Frobenius map on the Neron-Severi group of a supersingular abelian surface over a finite field?

Let $A$ be a supersingular abelian surface over a finite field $\mathbb{F}_q$. In that case the Neron-Severi group $NS(A\otimes\overline{\mathbb{F}_q})$ is the lattice of rank $6$. Are there methods ...
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4 votes
0 answers
245 views

What is the Artin invariant of an elliptic supersingular K3 surface?

Let $X$ be a supersingular K3 surface over an algebraically closed field $k$ of positive characteristic $\!p$. Artin proved in the paper https://eudml.org/doc/81948 that the determinant $\mathrm{disc}(...
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4 votes
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170 views

Is a Kummer surface unirational over a sufficiently large finite field of characteristic 2?

Let $A$ be a supersingular abelian surface over a sufficiently large finite field $\mathbb{F}_q$ of characteristic $2$ and let $K_A = A/(-1)$ be the Kummer surface. Shioda ("Kummer surfaces in ...
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4 votes
0 answers
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Obstruction to lifting of global sections of invertible sheaves

Are there known examples of smooth projective hypersurface in $\mathbb{P}^3$, say $X$ and an invertible sheaf $L$ on $X$ with $H^0(X,L)>0$ satisfying the following property: There exists an ...
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4 votes
0 answers
190 views

Blow-up of $\mathbb{P}^4$ along a smooth surface

Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$). In dimension ...
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3 votes
0 answers
186 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 ...
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3 votes
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129 views

Log canonical surface with an elliptic singularity

I would like to know if there is an example as follows: $X$ is a log canonical surface and $x \in X$ is an elliptic singularity such that The minimal resolution of $x$ is a circle of rational curves (...
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3 votes
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208 views

Historical proof of Leschetz Hyperplane Theorem

I browse in Phillip Griffiths' Slides on historical development of Hodge-theory and these include a sketch of the original approach with Lefschetz used to study complex surfaces in his famous ...
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3 votes
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161 views

Singular del Pezzo surfaces and degeneration of root systems

Let $S$ be a smooth del Pezzo surface of degree $d$ and $K_S^*$ the anticanonical class. It is well known that the set of classes $$R(S)=\{\alpha\in H^2(S,\mathbb Z)|\alpha^2=-2,\alpha\cdot K_S^*=0\},$...
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  • 1,013
3 votes
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"Simplification" of the map constructed at the proof of Castelnuovo's contractibility theorem

I'm reading the proof of the Castelnuovo's contractibility criterion in Beauville's book(Theorem II.17), and I guess I could understand all its affirmations. But I still has one question. For those ...
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3 votes
0 answers
233 views

Is the surface in $\mathbb{A}^3$ rational?

Consider the surface $$ (u_1^6 + 1)w^3 = (u_2^6 + 1) \subset \mathbb{A}^3 $$ over an algebraically closed field of characteristic $p \neq 2,3$. Is it rational, i.e., is there its proper ...
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3 votes
0 answers
187 views

Explicit equations for rational elliptic surfaces (Halphen surfaces)

I am looking for explicit equations for rational elliptic surfaces in characteristic $2$. For me, a rational elliptic surface $X$ is a smooth projective surface $X$ which is rational and equipped with ...
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3 votes
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138 views

Are there three ordinary elliptic curves $E$, $E_1$, $E_2$ such that $E^2 \cong E_1 \!\times\! E_2$?

Consider the elliptic curve $E\!: y^2 = x^3 + 1$ of $j$-invariant $0$ over an algebraically closed field $k$ of characteristics $p$. Let me remind that $E$ is ordinary (i.e., non-supersingular) iff $p ...
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3 votes
0 answers
123 views

Is the generalized Kummer threefold rational in characteristics 3?

Let $E_i\!: y_i^2 = x_i^3 - x_i$, $i = 1, 2, 3$ be three copies of the supersingular elliptic curve in characteristics $3$. Consider on $E_i$ the following automorphism of order $3$: $$ \sigma(x_i,...
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3 votes
0 answers
73 views

Under what conditions are superspecial abelian surfaces isomorphic over a finite field?

Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
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3 votes
0 answers
234 views

Are unirational K3 surfaces defined over finite fields?

Is every supersingular (thus unirational for ${\rm char }\ k = p\geq 5$, from Liedtke) $K3$ surface defined over a finite field? I guess this is true for Kummer surfaces, for example, since ...
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3 votes
0 answers
356 views

What is the most useful rationality criterion of surfaces?

The motivation for this question is that I would like to extract some information from derived category of surfaces to conclude the rationality of surface. There is a well known rationality criterion ...
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  • 1,516
3 votes
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249 views

How much information is encoded in the Jacobian-Kummer K3 surface of a curve of genus two?

Assume we work over $\mathbb{C}$. Let $S\subset \mathbb{P}^3$ be a quartic surfaces with 16 nodes (ordinary double points). Then there is a simple principally polarized abelian surface $(A,\theta)$ ...
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3 votes
0 answers
237 views

Intuition behind results in Mumford's “Lectures on curves on an algebraic surface”, II

NOTE: This is a followup to my question here. These are some questions concerning Mumford's "Lectures on curves on an algebraic surface". We concern ourselves with questions of the Picard variety $P$...
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