Birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where ...

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2answers
460 views

Relative minimality for conic bundles

Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces. The definition is ok: the fiber over ...
7
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1answer
428 views

Does a birational involution of C^n always have a fixed point?

The title describes completely the question. For n=1 it is an easy exercise. For n=2 the statement is still true, and depends on the classification of birational involutions of P^2 (see e.g. ...
10
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2answers
656 views

Rationality of GIT quotients

I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure: Every nonhyperelliptic genus 3 curve is a smooth plane ...
3
votes
3answers
293 views

Nefness of $h-e$ in the blowup of $\mathbb{P}^n$

Let $S$ be the blow up of $\mathbb{P}^n$ in a point $P$. Let $h$ be the class of the pullback of an hyperplane of $\mathbb{P}^n$ and $e$ the class of the exceptional divisor. Why is the divisor ...
2
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5answers
601 views

Reference for explicit calculation of blowups (of ideals) and strict transforms

Can one suggest some references where explicit calculations for blow up technique(along an ideal) and strict transformation is done in different examples?
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3answers
462 views

Why is the Hodge class of \bar{M_g} big and nef?

Let pi: \bar{Mg,1} \to \bar{M_g} be natural projection of compactified moduli stacks of curves and let omega be the relative dualizing sheaf. Then the Hodge class \lambda of \bar{M_g} is the first ...