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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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 ...
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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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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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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 ...
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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 ...
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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 $Q$: $$S_2\times Q\rightarrow Q,\; (\sigma,(x,y))\mapsto\... 3answers 971 views Divisor class group on blowup of nodal surface The following got no answer on mathstackexchange. I believe it not to be hard, but maybe it is a little specialized? All varieties will be over \mathbb{C} and projective unless stated otherwise. ... 1answer 723 views Automorphisms of del Pezzo surfaces Let S be a del Pezzo surface of degree six over \mathbb{C}. Then S is the blow-up of \mathbb{P}^2 in three general points p_1,p_2,p_3. Is it true that its automorphism group is ((\mathbb{C}... 3answers 438 views Is there a way to find any non-trivial \mathbb{F}_p(t)-point on the given elliptic curve? Consider a finite field \mathbb{F}_p (where p \equiv 1 \ (\mathrm{mod} \ 3), p \equiv 3 \ (\mathrm{mod} \ 4)) and the elliptic curve$$ E\!:y^2 = x^3 + (t^6 + 1)^2 $$over the univariate ... 1answer 838 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 ... 1answer 1k views 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 ... 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? 3answers 896 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 ... 1answer 411 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 ... 1answer 366 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 ... 1answer 258 views Field of definition for general type surfaces In the survey paper https://arxiv.org/abs/1004.2583 of Bauer-Catanese-Pignatelli, they mention a question of Mumford: Can a computer classify all surfaces of general type with p_g=0? I've been ... 0answers 183 views Map associated to linear system onto curve is morphism In Mumford's first paper on Surfaces in char p , 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, ... 1answer 4k views Algebraic equivalence VS Numerical Equivalence - An Example. This question is arose from the question Difference between equivalence relations on algebraic cycles and the example 3 in lecture 1 in Mumford's book Lectures on curves on an algebraic surface. ... 0answers 162 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 ... 1answer 165 views Is there a way to find any \mathbb{F}_2(t)-point on the elliptic curve \mathcal{E}? Consider the ordinary elliptic curves$$ E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1  over the field $\mathbb{F}_2$. They are quadratic twists to each other....
Let $S^{}$ 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 ...
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 ...