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.
677
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Is there a classification of minimal algebraic threefolds?
The minimal model program aims to find a minimal representative in the birational class of a given variety with reasonable singularities. Assuming this has been done, it seems natural to ask what ...
3
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0
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Birationally equivalent elliptic curves and singularities
I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3
\alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma
-\beta ^2$ for known ...
2
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1
answer
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Question regarding linear system of projective space
I am currently reading the paper titled "Birational Geometry of Moduli spaces of Configurations of Points on the Line" by M.Bolognesi and A.Massarenti. I have following doubts in section 2....
4
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Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?
Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...
6
votes
1
answer
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Different definition of Cox rings
Definition: Let $X$ be a normal projective variety with finitely generated Picard group. Define the Cox ring of $X$ as the multisection ring $$\text{Cox}(X)=\bigoplus_{(m_1,\ldots,m_k)\in \mathbb{N}^k}...
2
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1
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Sheaves on families of genus 2 curves in Hassett's paper
Sorry for a maybe stupid long question but I'm reading the paper "Classical and minimal models of the moduli space of curves of genus two" by Brendan Hassett and I'm not able to unravel a ...
5
votes
1
answer
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Equivalent definitions of Kodaira dimension
The Kodaira($-$Iitaka) dimension of a line bundle $L$ on a complex manifold $X$ can be defined either in three ways:
The maximal dimension of the image of the rational maps $φ_{|mL|} : X \...
1
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1
answer
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Positivity of the global log canonical threshold of a pair
Let $(X,L)$ be a polarized smooth projective variety. Let $D$ be a smooth irreducible divisor in $X$. Let $0<c<1$ be a real number. We denote $cD$ as $\Delta$. We can define the $\alpha$ ...
6
votes
1
answer
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Singular curves of genus 1
Let $C$ be an irreducible curve of arithmetic genus $1$ over a field $k$ and with a double $k$-point $p\in C$.
Is $C$ rational over $k$?
If $C$ is a plane cubic the answer is positive since we can ...
4
votes
1
answer
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Volume of conic bundles
Consider a smooth conic bundle $X\rightarrow \mathbb{P}^1$ with discriminant of degree $d$ (the locus of $\mathbb{P}^1$ over which the fibers are reducible conics). There is a formula for $(-K_X)^2$ ...
3
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1
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Confusion about the (Grothendieck–Poincaré) double dual of reflexive differentials vs usual differentials on a normal Cohen–Macaulay scheme
$\DeclareMathOperator\Hom{Hom}$Let $\mathcal{A}$ be an abelian category, my question is about the case when $\mathcal{A}$ is the category of quasi-coherent sheaves on a scheme $X$. There is a fully ...
4
votes
2
answers
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Smoothness of fibers over finite fields
Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties over a finite field of characteristic different from $2$. Is there any result on the existence of a point $y\in Y$ such that $X_y = ...
2
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0
answers
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Does anyone know a rationally chain connected, Cohen Macaulay variety which is not separably rationally connected?
An $ n $-dimensional variety (here variety means an integral, separated, scheme of finite type over an algebraically closed field) $ X $ over a field $ k $ is rational if there is a birational map $ \...
4
votes
1
answer
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Del Pezzo surfaces of degree four and complete intersections of two quadrics
Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$.
Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
6
votes
2
answers
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Smooth complete intersections
Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the ...
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0
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Why do such a birational map exists? And why it is unique?
Let $G$ be a complex linear algebraic group which is connected and reductive and let $\mathfrak g$ be its Lie algebra.
Suppose that $H \subset G$ is a 1-dimensional torus such that
the action of $H$ ...
1
vote
1
answer
162
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Space of rational conics
Let $K$ be a field of characteristic different from two. Conics over $K$ (that is curves of degree two in $\mathbb{P}^2_K$) are parametrized by $\mathbb{P}(k[x,y,z]_2) = \mathbb{P}^5_K$.
Conisider the ...
2
votes
1
answer
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Linear subspace in quadric hypersurfaces over a field
Let $K$ be a field of characteristic different from two, and $Q\subset\mathbb{P}^{n+1}_K$ an $n$-dimensional smooth quadric hypersurface over $K$.
Suppose also that $Q$ has a $K$-point and so $Q$ is ...
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Show that a polynomial of degree 4 is birational equivalent to a polynomial of degree 3 [duplicate]
Suppose that $f_{4}(x)$ is a polynomial of degree 4 with no multiple roots, $C$ is the curve defined by $y^{2}=f_{4}(x)$, I want to show that there is a polynomial $f_{3}(x)$ of degree 3 with no ...
1
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0
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Dominant rational map and linear systems
Let $X$ be a (smooth) projective variety of dimension $n$ and $L$ be an invertible sheaf on $X$ such that there is a linear subspace $V \subset H^0(L)$ of dimension $n+1$ such that the associated ...
5
votes
1
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Rational contraction and Proj of section ring
I am reading the paper "Mori Dream Spaces and GIT" by Hu and Keel.
https://arxiv.org/abs/math/0004017
I cannot understand the proof of Lemma 1.6 in it.
Let $X$ be a normal projective variety....
4
votes
1
answer
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When is the birational Torelli problem for CY threefolds true?
I am aware from Borisov, Căldăraru, Perry and Ottem, Rennemo that what is known as the birational Torelli problem is false in general for Calabi-Yau threefolds, but I would like to know if there are ...
1
vote
2
answers
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Embedding of a blow-up
In $\mathbb{P}^1\times\mathbb{P}^2$ take a general divisor $X$ of type $(0,2)$. Consider two general divisors $H_1,H_2$ of type $(2,1)$ and set $Y = X\cap H_1\cap H_2$.
Let $Z$ be the blow-up of $X$ ...
1
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0
answers
227
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Hironaka's construction for compact Kähler manifolds
In Hartshorne's book 《Algebraic Geometry》 p.443, the author introduces a construction of a non-projective complex manifold from a projective one. His method can be summarized as following:
Let $X$ be ...
5
votes
2
answers
548
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Birational geometry over finite fields
I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
3
votes
1
answer
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Embedding quadric bundles
Let $\pi:X\rightarrow W$ be a morphism of smooth projective varieties over a field $k$ whose generic fiber is a smooth quadric, and let $r$ be the dimension of the fibers of $\pi$.
Does there always ...
2
votes
0
answers
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Rational points on surfaces
Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form
$$
S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\}
$$
where $...
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0
answers
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Questions about Hironaka's example
In Hartshorne's book 《Algebraic geomery》 p.443, the author gives an explanation of Hironaka's example on non-Kähler deformation of compact Kähler manifolds, his construction can be summarised as ...
4
votes
2
answers
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Are Du Val singularities smoothable?
I am interested in when a Du Val surface singularity is smoothable. By Du Val singularity, I mean (the germ of a) isolated double point surface singularity admitting a resolution by blowups of ...
7
votes
0
answers
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Number of rational points over finite fields mod $q$ is birational invariant
I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...
2
votes
1
answer
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Existence of terminal $3$-fold flips
Does there exist a terminal $3$-fold $X$ with a curve $C\subset X$ such that $K_X\cdot C < 0$ admitting a Mori flip $X\dashrightarrow Y$, flipping $C$ to a curve $C'\subset Y$, where the singular ...
5
votes
1
answer
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diagonal cubic hypersurfaces
At the end of
https://encyclopediaofmath.org/index.php?title=Cubic_hypersurface#References
it is stated that the diagonal cubic hypersurface
$$
\sum_{i=0}^{2m+1} a_i x_i^3 = 0, m\ge 2
$$
(and ...
4
votes
0
answers
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Parameter spaces for conic bundles
A conic bundle over $\mathbb{P}^n$ is a morphism $\pi:X\rightarrow\mathbb{P}^n$ with fibers isomorphic to plane conics. A conic bundle $\pi:X\rightarrow\mathbb{P}^n$ is minimal if it has relative ...
1
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0
answers
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What's the modification of a Calabi-Yau manifold?
Recall that a modification of a compact manifold $X$ is a holomorphic map $\mu:\tilde X \to X$ such that:
i) dim $\tilde X$=dim $X$;
ii) there exists an analytic subset $S\subset X$ of codimension $\...
9
votes
0
answers
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How can I "see" that a map is birational?
This came up with the Euler brick.
Let $T=(p,q,r)$ be a Randall triple, i.e. $$(p^2-1)(q^2-1)(r^2-1)=8pqr\ \qquad\text{[eq.1]}.$$ There are tons of maps that map a triple $T$ to another $T'=(p',q',r')$...
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1
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Does a resolution of a rational singularity have rationally connected fibers?
A rational singularity is a singularity of a
complex variety $X$ such that for any
resolution $\pi:\; \tilde X\rightarrow X$ the
higher direct images $R^i\pi_*(O_{\tilde X})$
vanish for all $i>0$. ...
4
votes
0
answers
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Minimal $b_2$ in Sarkisov's construction
In the paper On the structure of conic bundles. Math. USSR, Izv.,
120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}...
3
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1
answer
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Is the automorphism group of a normal affine scheme a group scheme or an algebraic space?
If $ \operatorname{Spec}(A) $ is a smooth affine scheme over an algebraically closed field $ k $, then is $ \operatorname{Aut}(\operatorname{Spec}(A)) $ a group scheme or an algebraic space?
Please ...
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0
answers
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Mori cone of Picard rank two varieties
Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that
$$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$
is an isomorphism, where $i:S\...
1
vote
1
answer
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Moduli spaces of horizontal curves
Let $f:X\rightarrow Y$ be a morphism of projective varieties. We may assume that $X$ and $Y$ are smooth, and $f$ is flat of relative dimension one. Fix an ample divisor $A$ on $X$.
I would like to ask ...
3
votes
0
answers
199
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Dynamical degree and spectral radius
Let $X$ be a smooth, projective surface over an algebraically closed field $k$ of characteristic zero, and let $f \in \mathrm{Bir}(X)$ a birational map.
Let's denote $f_{\ast} : \mathrm{NS}(X) \...
2
votes
1
answer
124
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Cremona transformations and divisors
Let $L$ be an ample line bundle in $\mathbb{P}^n$, with at least $n$ global sections. Choose two sets of $n$ linearly independent global sections of $L$, say $S_1:=\{D_1,...,D_n\}$ and $S_2:=\{E_1,.......
2
votes
1
answer
408
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Divisors on projective bundles
Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$.
If $n = 0$ then $X = \mathbb{P}^1$, and for $n ...
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Coefficients of elliptic curves over function fields
Consider the projective plane $\mathbb{P}^2_{\overline{\mathbb{C}(t)}}$ over the algebraic closure of the function field $\mathbb{C}(t)$.
Take the point $p_0 = [0:1:0]\in \mathbb{P}^2_{\overline{\...
3
votes
0
answers
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Does every Fano variety contain every abstract curve?
It is a famous result of Mori that all Fano varieties (in characteristic $0$) contain rational curves. What if we replace rational curve with a specific curve of positive genus?
Question. Is it true ...
2
votes
0
answers
401
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On the exceptional divisor of the resolution of indeterminacy locus of rational map
Let $f:X \dashrightarrow Y$ be a rational map between smooth, projective varieties over $\mathbb{C}$. We know that there is a resolution of the indeterminacy locus using which we obtain a smooth, ...
3
votes
1
answer
252
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On tensor product of field extensions
Let $K$ be a field which is a (transcendental) extension of $\mathbb{C}$. Let $L_1, L_2$ and $M_1, M_2$ be two field extensions of $K$ (not necessarily algebraic) such that $$L_1 \otimes_K L_2 \cong ...
2
votes
1
answer
232
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Blow-ups of surfaces over a field
Let $S$ be a smooth projective surface of Picard rank $\rho(S)$ over a field $K$, and $\overline{S}$ its algebraic closure.
Take a point $p\in\overline{S}$ and denote by $\overline{X}$ be blow-up of $\...
5
votes
2
answers
498
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Divisors whose restriction is big
Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$.
...
2
votes
1
answer
276
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Construction of Jacobian Ideal
In Qing Liu's Algebraic Geometry and Arithmetic Curve, we have the following proposition(6.3.13):
Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be smooth schemes over $S$. Then any immersion $f:...