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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264 views

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 ...
14
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2k views

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 ...
12
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196 views

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 ...
11
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277 views

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 ...
11
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441 views

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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173 views

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, ...
6
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146 views

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 ...
6
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156 views

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 ...
6
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120 views

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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189 views

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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592 views

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+\...
5
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190 views

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' \...
5
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162 views

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, ...
5
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206 views

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 ...
5
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152 views

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-...
5
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301 views

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 ...
5
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261 views

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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274 views

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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104 views

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 ...
4
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72 views

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 ...
4
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187 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 ...
4
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162 views

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 ...
4
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104 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 ...
4
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134 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 $y$ ...
4
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156 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 ...
4
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160 views

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^...
4
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218 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 ...
4
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161 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 ...
4
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141 views

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 ...
4
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221 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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165 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 ...
4
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188 views

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 ...
4
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0answers
187 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 ...
3
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222 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 ...
3
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149 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 ...
3
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135 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 ...
3
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110 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,...
3
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71 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 ...
3
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0answers
191 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 ...
3
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0answers
312 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 ...
3
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230 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 $(A,\theta)$ ...
3
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227 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 $P$...
3
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395 views

The Jacobian surface of an elliptic surface

Let $\mathcal{X}$ be an elliptic surface over $\mathbb{P}^1$ without a section and let $\mathcal{J}$ be an elliptic surface over $\mathbb{P}^1$ with a section. Assume we have the commutative diagram \...
3
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196 views

What is the integral cohomology of an Enriques surface over a finite field?

Probably this is well-known, but I could not find it. I would like to understand the integral $2$-adic etale cohomology of an Enriques surface over $\mathbb{F}_q$ in dimension 2: $H_{et}^2(X, \mathbb{...
3
votes
0answers
156 views

Elliptic fibration arising from a higher genus linear system

Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$. Let $L\subset H$ be a ...
3
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0answers
458 views

Does the Bombieri-Lang conjecture imply severe restrictions on rational points on twists of hyperelliptic curves?

According to Silverman, the Bombieri-Lang conjecture implies that the rational points of surface on general type lie on finite set of curves, except for a finite set of points. Let $f$ be univariate ...
3
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0answers
177 views

Pic^0 of the surface of bitangents of a quartic

Let $S$ be a generic quartic surface in $\mathbf{P}^3$. Let $T$ be the surface of the lines bitangent to $S$. What can we say about $Pic^0(T)$?
3
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264 views

Hypersurface with singularities

I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?). It was said ...
3
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0answers
328 views

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....