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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1answer
1k views

When do 27 lines lie on a cubic surface?

Consider $27$ (pairwise distinct!) lines in $\mathbb{P}^3$ whose intersection graph is that expected¹ of the $27$ lines on a smooth cubic surface. Question: Is there a simple necessary and sufficient ...
3
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3answers
876 views

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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2answers
739 views

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 ...
15
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1answer
3k views

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...
19
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3answers
3k views

Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$

I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle ...
6
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1answer
818 views

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

These are some questions concerning Mumford's "Lectures on curves on an algebraic surface". We concern ourselves with questions of the Picard variety $P$, and its dimension, of a complete nonsingular ...
10
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1answer
751 views

Restriction of the Picard group of a surface to a curve

In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious: For a general (smooth) surface $S$ in $\mathbb{P}^3$ ...
14
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4answers
3k views

Rational curves on varieties of general type

Let $S$ be a complex surface of general type. Are there infinitely many smooth rational curves on $S$? And more general, what if $V$ is a variety of general type?
7
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1answer
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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 ...
5
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3answers
917 views

Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$

Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$: $$S_2\times Q\rightarrow Q,\; (\sigma,(x,y))\mapsto\...
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3answers
872 views

Names of certain surfaces

Are there any generally used names for the following algebraic and nonalgebraic surfaces? Any references to literature where the surfaces are studied are also appreciated. Surface I. Implicit ...
8
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1answer
357 views

Endomorphism algebras of abelian surfaces with real multiplication

Given an abelian variety $A$ over a field $F$, one may consider the ring of endomorphisms $End(A)$, the ring of $F$-rational maps $A \to A$ respecting the group structure on $A$. We may also consider ...
7
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1answer
350 views

Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...
5
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0answers
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, ...
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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 ...
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152 views

Computing Euler Charactistics of Line bundles on Hilbert Schemes of points on Surfaces

Let $S^{[2]}$ be the Hilbert scheme of two points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...
2
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3answers
291 views

Linear homogenous polynomials that generates one quadratic polynomial

Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$. Assume that for every $i$ and ...