# Tag Info

### Rational curves on varieties of general type

I am surprised nobody mentioned the result of Lu and Miyaoka (Math. Res. Letter 2, 663-676 (1995)) which implies indeed that there are only finitely many smooth rational curves on a surface of general ...
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### Algebraic surfaces with no deformations

There are several different notions of "rigidity" (local rigidity, global rigidity, infinitesimal rigidity, étale rigidity and strong rigidity) and it is possible to provide examples for each of them. ...
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### A diffeomorphism of complex surfaces mapping subvarieties to subvarieties

I am posting this answer because the following linear algebra proposition is too long for a comment. Lemma. Let $A$, respectively $B$, be an invertible $\mathbb{R}$-linear operator on $\mathbb{C}^2$ ...
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### Topology change induced by small perturbation

The formal statement you are thinking of when you assert "The topology changes only when..." is Ehresmann's theorem: a proper smooth submersion is a fiber bundle, and hence all fibers are ...
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### Pull-back of an irreducible ample divisor via an isogeny of abelian varieties

I think the Proposition is not true if $D$ is singular. Take a smooth curve $C$ of genus 2, and $X=JC$; embed $C$ in $X$ (say, by choosing a point of $C$). Let $\alpha$ be a point of order 2 in $X$; ...
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### Intersection graph of $(-1)$-class divisors on surface of general type

On any surface $X$ with non-negative Kodaira dimension the $(-1)$-curves (i.e, the smooth rational curves $D$ with $D^2=-1$) are isolated, in other words any two of them do not intersect. From this ...
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### Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?

You can observe that the elliptic curve $E_2: y^2=x^3+(t^3+1)^2$ is a generic fibre of a rational elliptic surface. Over the algebraically closed field $k$ the group of $k(t)$ points on the curve has ...
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### Are any of these complex surfaces ever projective?

Here is a simple method for constructing projective examples: Assume there exist maps $f:C \to \mathbb{P}^1$ and $g:T \to \mathbb{P}^1$ of the same degree which are totally ramified at $c$ and $t$. ...
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### Motivation for birational geometry

what are some interesting properties of varieties that are preserved under birational transforms? I will answer the question for smooth projective varieties (certainly a geometrically nice class of ...
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### Motivation for birational geometry

Will has already said many of the things that I would have said trying to answer your extended question, but let me add a few things without trying to avoid overlap. In fact, let me start with an ...
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### What is the surface area of the finite part of the Cayley nodal cubic surface?

Here is the start of an answer - I am afraid that I don't want to spend any more of my time thinking about this problem, but it gets the answer in terms of an integral that looks very computable. ...
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### General conditions for normality of blow-up

Let $X=Spec(R)$. Blowing-up $Z=V(I)$ is the same as to look at $Proj$ of the graded ring $R[It]=\oplus_{j\geqslant 0} I^jt^j\subset R[t]$, the Rees ring associated to $I$. Assume $R$ is a domain, ...
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### Volume of a divisor on a smooth projective surface

At least for effective divisors, the answer is strongly related to Zariski decomposition. If $D$ is an effective divisor on a smooth surface $X$, Zariski proved in [Z62] that there exists a unique ...
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### Seeking concrete examples of "generic" elliptic fibrations of K3 surfaces

As far as I know no one has “written down” a hyperkähler structure on a K3 surface. Indeed much of the work in Mirror symmetry revolves around trying to give an asymptotic expansion of such metrics (...
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### Multiplicity of irreducible component of a singular fiber of a $\mathbb{P}^1$-fibration
If we assume that the rational curves intersect in nodes, then we can compute it from the data of the dual graph. Since $L$ intersects only one of the $D_i$, then the $D_i$ necessarily form a chain, ...