Questions tagged [birational-geometry]
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.
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Negative definite of exceptional curve in higher dimension
One direction of the Grauert's contractibility theorem shows
Let $f:S\rightarrow T$ be a surjective holomophic map where $S$ is a compact holomorphic surface. If $C$ is a reduced connected effective ...
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Seeking for bridges to connect K-stability and GIT-stability
We consider the variety $\Sigma_{m}$ := {($p$, $X$) : $X$ is a degree $n$ + 1 hypersurface over $\mathbb{C}$ with mult$_{p}(X) \geq m$} $\subseteq$ $\mathbb{P}$$^{n}$ $\times$ $\mathbb{P}$$^{N}$, ...
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Birational deformations of holomorphic symplectic manifolds
Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\...
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Behavior of divisors under push forward and pull back
Consider a birational morphism between smooth projective varieties $f:X\to Y$. I would like to understand the behavior of push-pull/pull-push of effective divisors under $f$. I know that if $D$ is an ...
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Flops connect minimal models of algebraic spaces?
According to a Kawamata's result, two projective minimal models of the same variety are connected through a sequence of flops. In particular, a birational map $f\colon X\to X'$ between Calabi-Yau ...
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Geometry of destabilizing centers in $K$-stability
In $K$-stability destabilizing centers are, roughly speaking, centers of valuations computing the stability thresholds.
It is known that if $X$ is non $K$-semistable Fano variety then there exists a ...
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Quotient of the plane by the standard Cremona involution
Consider the standard Cremona involution $i:\mathbb{P}^2\dashrightarrow \mathbb{P}^2$, $[x:y:z]\rightarrow [yz:xz:xy]$.
Let $Y$ be the blow-up of $\mathbb{P}^2$ in the three base points of $i$, so ...
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Is there some criterion for a divisor in the restricted base locus will always be contracted by the minimal model program?
Let $(X,B)$ be a KLT pair, which is of log general type, that we can run MMP with scaling $H$ by BCHM and gets $\phi:(X,B)\dashrightarrow (X^m,B^m)$.
The question is: Is there some criterion that the ...
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Nef cone of Hilbert scheme of $n$ points
Suppose $\operatorname{Nef}(X)$ is a rational polyhedron with extremal rays $\{F_i\}_i$. Now, consider the Hilbert scheme of $n$ points $X^{[n]}$ and the embedding $\operatorname{Nef}(X)\subset \...
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A proof in BCHM about the finite generation of the Cox ring
I was reading the paper Birkar, Cascini, Hacon, and McKernan - Existence of minimal models for varieties of log general type.
There is a consequence about a finite generation for Cox ring associated ...
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Is every normalization a blowup?
I asked this at math.stackexchange, but received no reply.
Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example.
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Embedding degree 1 Del Pezzo surfaces in $\mathbb{P}(1,1,2,3)$
In the projective bundle $\mathbb{P}(\mathcal{O}(-1)\oplus \mathcal{O}(-1)\oplus \mathcal{O})\rightarrow\mathbb{P}^1$ consider the hypersruface
$$
X := \{a_{00}y_0^2+a_{01}y_0y_1+a_{02}y_0y_2+a_{11}...
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Terminal singularities of fibers vs total space
Suppose $f\colon X \to Y$ is a flat map of complex varieties (or more generally DM stacks?). Suppose every fiber has at most terminal singularities and that $Y$ is smooth. Under what conditions is it ...
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Exceptional locus of rational map has codimension two
Let $f: X\dashrightarrow\mathbb{CP}^n$ be a birational map where $X$ is a smooth, projective variety. Then there are some closed subvarieties $Z'\subset X$ and $Z \subset \mathbb{CP}^n$ such that $f$ ...
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Contraction of $(-1)$ curve and extremal ray
I want to prove Castelnuovo's contraction theorem by Mori's contraction theorem.
Question. How can one show that a $(-1)$ curve on a smooth surface is an extremal ray?
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