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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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 $X_A := f^{-1}(A)$.
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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:...
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0
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Fibers of period map
Consider the period map for some smooth varieties, $p:\mathcal{X}\rightarrow\mathcal{A}:X\mapsto J(X)$, where $\mathcal{X}$ is moduli space of certain varieties and $\mathcal{A}$ is moduli space of ...
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How to distinguish the singularities on moduli space?
Let me start with concrete examples. Let $X$ be a smooth special Gushel-Mukai threefold and $\mathcal{C}(X)$ be its honest Fano surface of conics, it has two irreducible components $\mathcal{C}(X)=\...
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Curves in conic bundles
Consider a smooth minimal $3$-fold conic bundle $f:X\rightarrow\mathbb{P}^2$. Then $X$ has Picard rank two and consequently also the vector space of $1$-cycles is $2$-dimensional. Then the cone of ...
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Motivation for birational geometry
I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
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Simultaneous Log resolutions for both varieties and divisors
Let $X$ be a normal variety and $D \subset X$ be a prime divisor which is also normal. It is well-known that we can take a resolution $f: W \to X$ of $X$ such that
$$\DeclareMathOperator{\Supp}{\...
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Global sections of multiples of a divisor
Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$.
Is there any example where $h^0(X,mD) = 0$ ...
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A question on effective divisors
Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...
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Nef and pseudo-effective divisors over non algebraically closed fields
Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective.
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On the b-nefness of the moduli part of canonical bundle formula
I have recently been wondering about the existence of a canonical bundle formula in the following situation and am not sure how to proceed.
Suppose $(X,B) \xrightarrow{f} Y$ is a fibration where $ (X,...
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Canonical covering stack of a flop
In section 6 of Kawamata's paper https://arxiv.org/abs/math/0205287, he defines the canonical covering stack. In the proof of theorem 6.5, he considered a flop $$X\xrightarrow{\phi}W\xleftarrow{\psi}Y$...
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Pseudoeffective divisors on surfaces
Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $...
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Examples of complex manifolds with trivial Néron–Severi group?
$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$Let $X$ be a compact complex manifold, assume projective if you'd like. Define the Néron–Severi group to be the quotient $$\NS(X) = \Pic(X) / \...
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Log canonical surface with an elliptic singularity
I would like to know if there is an example as follows:
$X$ is a log canonical surface and $x \in X$ is an elliptic singularity such that
The minimal resolution of $x$ is a circle of rational curves (...
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Restriction of small transformations
Let $\phi:X\dashrightarrow Y$ be an elementary small transformation (isomorphism in codimension $1$) between normal and $\mathbb{Q}$-factorial projective varieties.
Then there are small contractions $...
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descent of nef divisors
Let $f:X \rightarrow Y$ be a flat surjective morphism of complex projective varieties with connected fibers, $X$ normal, $Y$ smooth and $dim (Y) < dim(X)$. Suppose $L \in Pic (X)$ such that $L|_{X_{...
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Rationality in pencil of projective varieties
Let $\pi: \mathcal{X} \to \mathbb{P}^1$ be a pencil of projective $\mathbb{C}$-varieties such that a general fiber is smooth. Let $\mathbf{P}$ be one of the properties: rational, unirational, stably ...
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Rational and rationally chain connected surfaces
A projective variety $X$ over the complex numbers is rationally connected if two general points of $X$ can be joined by a rational curve in $X$, and rationally chain connected if two general points of ...
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Demailly Campana Peternell Conjecture for isolated singularities
I have asked this question on StackExchange but didn't get an answer, therefore I am asking again here.
If $M$ is smooth, and $T^*M\to$ Spec$H^0(T^*M,\mathcal{O}(T^*M))$ is a projective birational map,...
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Subsets of $\mathbb{N}$ arising as genera of smooth projective curves in a variety
Given a smooth projective variety, the genera of the smooth projective curves in it form a subset of $\mathbb{N}$.
Assuming the dimension is at least $2$, I think this subset is polynomially spaced, i....
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Cone and contraction theorems for certain sub-klt pairs
Suppose $(Y,B_Y)$ is a sub-klt pair which is a birational model of a klt pair $(X,B)$: there exists a birational morphism $\pi:(Y,B_Y) \rightarrow (X,B)$ such that $\pi^{*}(K_X+B)=K_Y+B_Y$. We know ...
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Identifying plane scrolls
In this paper it is shown (Corollary 1.9) that if for a 3-dimensional variety $X\subset \mathbb{P}^r$ there is an open set of hyperplanes $Y\in(\mathbb{P}^r)^{\vee}$ such that $\forall H, H'\in Y$ the ...
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An invariance property of rational singularities
Let $X$ be a normal variety over a field of characteristic zero with rational singularities.
If $\pi:Y \to X$ is a birational proper morphism with $Y$ also normal, then does $Y$ also have rational ...
3
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Pullback of $\mathbb{R}$-Cartier divisors
I am reading the recent book by Kawamata, Algebraic Varieties: Minimal Models and Finite Generation. There is an English translation here .
In the bottom of page 16 he says that an $\mathbb{R}$-...
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Domain of definition of a rational map is determined by several coherent sheaves of ideals
I am reading the Hironaka's paper on desingularisation. One part of his work is about eliminating the indeterminacy of proper rational map. He says that this comes from the Principalization of Ideal ...
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1
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A question on linear projection of a smooth projective variety
Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $n$. Fix a closed point $x \in X$ and an embedding of $X$ in $\mathbb{P}^m$ for some integer $m$. For a given $d$, denote by $\sigma_d ...
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Algebraic closure of $\mathbb{C}(t)$
Let $\mathbb{C}(t)$ be the field of rational functions $f(t) = \frac{p(t)}{q(t)}$ with $p,q\in\mathbb{C}[t]$.
For instance, the function $g(t) = \sqrt{t}$ does not belong to $\mathbb{C}(t)$ but is ...
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Cohomology of finite birational morphism
Let $f:Y \to X$ be a finite birational morphism between projective varieties (over $\mathbb{C}$) with $Y$ non-singular. I want to understand the cokernel of the pull-back morphism from $H^q(X,\mathbb{...
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Moving general fibers of a fibration
Let $X$ be an irreducible projective variety over $\mathbb{C}$ admitting a morphism $\pi:X\rightarrow \mathbb{P}^1$ with connected fibers. We may assume that the general fiber of $\pi$ is smooth.
My ...
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Mori fiber space in dimension $2$ over a point is $\mathbf P^2$
Let $k$ be an algebraically closed field. Let $X$ be a surface over $k$. Let $\pi: X \to S$ be an extremal contraction. It is well-known that if $\dim(S) = 0$, then $X \cong \mathbf P^2$. I wonder if ...
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Irreducibility of the base and of the general fiber
Let $f:X\rightarrow Y$ be a morphism of scheme over $\mathbb{C}$. Assume that $Y$ and the the general fiber $F_y = f^{-1}(y)$ of $f$ are irreducible.
Does there exists an irreducible component $X'$ of ...
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Question about valuation and blow up (a lemma in GIT book)
I'm reading Mumford's book Geometric Invariant Theory and confused about the proof of a lemma on Page 91&92:
Lemma:
Let $V_0$ be a smooth surface over an algebraically closed field $k$
with char$...
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1
answer
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Moduli spaces and conic bundles
The moduli space $A_2(1,8)^{\operatorname{lev}}$ of $(1,8)$-polarized abelian surfaces with canonical level
structure has a structure of conic bundle over $\mathbb{P}^2$ with a curve of degree $4$ as ...
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1
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Set theoretic equation for Veronese varieties
Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...
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0
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Rational varieties over finite fields admit an open set isomorphic to an affine space
This paper roughly claims that given a projective variety $X$ over a finite field, there is a finite map $f:X\rightarrow \mathbb{P}^n$ such that if $H$ is a hyperplane in $\mathbb{P}^n$ and $U=\mathbb{...
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3-secant lines of a projective curve
Consider a smooth projective curve $C\subset\mathbb{P}^n$. Let $G(1,n)$ the Grassmannian of lines of $\mathbb{P}^n$. The variety $S_2(C)\subset G(1,n)$ parametrizing lines that are secant to $C$ (i.e.,...
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0
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Factorization of birational maps in char $p$
So I was reading about the factorization result that any birational map between smooth varieties is composition of blow-ups and blow-downs with smooth centers. It is apparently true only in ...
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How many characteristics is a random surface unirational in?
Suppose I have a surface $X$ defined over $\mathbb{Z}$. I am interested in the set $S_X$ of primes $p$ such that $X_{\overline{\mathbb{F}}_p}$ is unirational. If I choose a "random" surface ...
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Does self-product of universal hypersurfaces have dlt singularites?
Let $n\geq2,d\geq 2n+1$ be integers, let $\mathcal{X}_{n,d}\to|\mathcal{O}_{\mathbb{P}^n}(d)|$ be the universal family of hypersurfaces. The total space $\mathcal{X}_{n,d}$ is smooth as it is a ...
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Picard groups of determinantal varieties
Consider a general $4\times 4$ matrix:
$$
X:=\left(
\begin{array}{cccc}
X_0 & X_1 & X_2 & X_3 \\
X_4 & X_5 & X_6 & X_7 \\
X_8 & X_9 & X_{10} & X_{11} \\
X_{12} &...
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Linear spaces secant to Veronese varieties
The following question makes sense in a more general setting but for sake of simplicity let me stick to a particular case.
Consider the degree three Veronese embedding $V\subset\mathbb{P}^9$ of $\...
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1
answer
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Termination of a minimal model program
I am reading "The dual complex of
singularities" by de Fernex, Kollár
and Xu and in the proof of Corollary 24 I have encountered a bit of
reasoning that confuses me.
Let $(X, \Delta)$ be a $\...
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2
answers
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Picard group of a cubic hypersurface
Consider the following cubic hypersurface in $\mathbb{P}^5$:
$$
X = \{z_0z_3z_5-z_1^2z_5-z_0z_4^2+2z_1z_2z_4-z_2^2z_3 = 0\}\subset\mathbb{P}^5
$$
The singular locus of $X$ is the Veronese surface $V\...
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Is the surface in $\mathbb{A}^3$ rational?
Consider the surface
$$
(u_1^6 + 1)w^3 = (u_2^6 + 1) \subset \mathbb{A}^3
$$
over an algebraically closed field of characteristic $p \neq 2,3$. Is it rational, i.e., is there its proper ...
2
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0
answers
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Divisorial contraction to a non-normal variety
Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which ...
5
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1
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Isomorphisms of complete intersections
Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d_1,…,d_r)$.
If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so ...
3
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1
answer
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Weak Fano varieties and small transformations
A projective normal and $\mathbb{Q}$-factorial variety $X$ is said to be log Fano if there exists and effective divisor $D$ on $X$ such $-K_X-D$ is ample and the pair $(X,D)$ is klt.
Now, let $f:X\...
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vote
1
answer
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Push-forward of divisors and intersections
Let $f:X\rightarrow Y$ be a surjective finite morphism of varieties, with $X$ normal and $Y$ smooth. Let $D\subset X$ be a divisor and $C\subset Y$ a curve. Does the equality
$$C\cdot f_{*}D = f^{*}C\...
2
votes
1
answer
307
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Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold
Let $X$ be a smooth general ordinary Gushel-Mukai threefold. There is an embedding $X\rightarrow\mathrm{Gr}(2,5):=G$. Consider the normal bundle $\mathcal{N}_{X|G}$, how to compute cohomology of this ...