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Results tagged with birational-geometry
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user 5101
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 the rational functions have poles.
3
votes
When is a del Pezzo surface a conic bundle?
It is not clear from the question whether you want to determine whether $X$ admits a conic bundle structure, or whether it is birational to a conic bundle surface. I will focus on the former. If you a …
- 21.4k
2
votes
1
answer
190
views
A variant on the Fujita invariant
Let $X$ be a Fano variety over $\mathbb{C}$. Let $D$ be a divisor on $X$. Recall that the Fujita invariant of $D$ is defined to be
$$a(D) = \inf \{ t \in \mathbb{R} : K_X + tD \text{ is effective} \}. …
- 998
4
votes
0
answers
200
views
Models of conic bundles
Let $S$ be a smooth projective variety over $\mathbb{C}$. A conic bundle over $S$ is a smooth projective variety $X$ together with a flat morphism $\pi:X \to S$ all of whose fibres are isomorphic to p …
- 21.4k
3
votes
Space of rational conics
Let $$X: \quad a_{0,0}x^2 + a_{1,1}y^2 + a_{2,2}z^2 + a_{0,1}xy + a_{0,2}xz + a_{1,2}yz = 0 \quad \subset \mathbb{P}^2 \times \mathbb{P}^5$$
be the total space of the family of all plane conics. Then …
- 21.4k
5
votes
A property of varieties between unirational and retract rational
Let $k$ be a field of characteristic $0$, $a \in k$ and $f$ a separable polynomial of degree $3$.
The projective surface $X$, given as the minimal smooth compactification of the affine surface
$$X: …
- 21.4k
5
votes
Accepted
Non-rational, smooth Fano 3-folds with $\rho(X) >1$
A complete list of non-rational Fano threefolds, together with results towards non-stable rationality, can be found in the paper:
BRENDAN HASSETT AND YURI TSCHINKEL- ON STABLE RATIONALITY OF FANO THR …
- 21.4k
7
votes
Accepted
Obstruction to rationality of del Pezzo surfaces of degree 4
Interesting question.
But, alas, the answer is no.
The issue is that you have missed an extra non-rationality criterion. Namely, it is possible that such a surface $X$ has $\mathrm{Br}(X) = \mathrm{ …
- 21.4k