Skip to main content

Questions tagged [birational-geometry]

For questions on birational geometry, a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets.

Filter by
Sorted by
Tagged with
0 votes
1 answer
38 views

A line bundle that is relative big over a Zariski dense open subset is also big on every fiber

Let $f:X\to S$ be a proper morphism, let $L$ be a line bundle defined over $X$ which is relative big over $S^0$ where $S^0 \subset S$ is a Zariski dense open subset, prove $L$ is also relative big ...
yi li's user avatar
  • 4,828
3 votes
0 answers
73 views

Is every normalization a blowup?

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. Let $Y \to X$ be the normalization. The answer is positive in ...
SeparatedScheme's user avatar
0 votes
1 answer
50 views

Isomorphism in codimension $1$ and Cartier divisor

We work over $\mathbb{C}$. Let $X,Y$ be normal projective varieties, and let $f: X\dashrightarrow Y$ be a birational map among them. Assume that $f$ is an isomorphism in codimension $1$, that is the ...
ark's user avatar
  • 47
0 votes
1 answer
44 views

Does a terminal Gorenstein cDV singularity imply a DuVal singularity on the general elephant?

I am reading through Kollár and Mori's book "Birational geometry of algebraic varieties". Now I was trying to understand Theorem 5.43, I only state the relevant part for the question. Let $(...
AOJIDSOeoi's user avatar
0 votes
0 answers
29 views

Cones of divisors on $\text{CDiv}(X)_{\mathbb{R}}$

Let $X$ be a normal, projective $\mathbb{Q}$-factorial variety over the complex numbers. I know that, in order to understand the birational geometry of $X$, one may consider several cones of divisors, ...
ark's user avatar
  • 47
1 vote
0 answers
30 views

Morphism from a minimal model to the canonical model

Assume that $(X,B)$ is a lc/klt pair where $B$ is an $\mathbb{R}$-divisor, and $(Y,B_Y)$ is a minimal model of $(X,B)$ such that $K_Y+B_Y$ is semi-ample, i.e. there is a contraction morphism $g:Y\...
Hobo's user avatar
  • 327
2 votes
1 answer
50 views

Why does the MMP terminates when the canonical divisor is nef?

I have recently taken interest in Mori's Minimal Model Program (MMP) and I struggle to figure out why it stops when the canonical divisor $K_X$ of our variety $X$ is nef. For now, I have understood ...
user1319604's user avatar
1 vote
0 answers
61 views

Why the "log" in "log resolution"?

I have learned about log resolutions and log canonical thresholds. Recently, I wondered what the "log" in this terms means and where it comes from. I suspect, it is related to the usual ...
Daniel W.'s user avatar
  • 1,788
0 votes
0 answers
35 views

How much can we control a birational locus for Chow’s lemma

Let $X$ be a smooth proper complex variety. By Chow’s lemma, there exists a surjective birational morphism $X’\to X$ with $X’$ projective. My question is the following: take a closed point $x \in X$. ...
P. Usada's user avatar
  • 400
5 votes
1 answer
62 views

Is the curve $y^7=x^2(x-1)^2$ hyperelliptic?

When playing around with genus $3$ curve $X$ which has an automorphism $\sigma$ of order $7$. Using the Riemann-Hurwitz formula, one find the map $X\to Y:=X/\langle\sigma\rangle$ is a degree $7$ map ...
cybcat's user avatar
  • 481
0 votes
0 answers
40 views

rational map by F. Mangolte

I'm reading Real Algebraic Varieties by F. Mangolte. Definition 1.3.22 (in the book) If $X$ and $Y$ are algebraic varieties over a base field $K$ a rational map $\phi:X\dashrightarrow Y$ is an ...
isz's user avatar
  • 31
0 votes
0 answers
24 views

An Example of Flipping Contraction: the Contraction of Zero Sections of Vector Bundles?

Here is an example of flipping contraction: Consider the projective space $\mathbb{P}^d$ with vector bundle $\mathscr{E}:=\mathscr{O}(-a_1)\oplus\cdots\oplus\mathscr{O}(-a_r)$ for $a_i>0$. ...
WakeUp-X.Liu's user avatar
2 votes
1 answer
54 views

When the image of flipping and flipped loci of a dlt pair are the same?

I'm learning the birational geometry and I have the following elementary question but it seems no books talk about it: Let $(X,B)$ be a DLT pair (with $\mathbb{R}$-coefficients), assume we have a ...
WakeUp-X.Liu's user avatar
1 vote
0 answers
55 views

A question from Fujino's book

In Fujino's book Foundations of the minimal model program, the definition of semi-ample $\mathbb{R}$-divisors is as follows Then he claims that I don't understand the last sentence of the proof. Why ...
Hobo's user avatar
  • 327
1 vote
1 answer
67 views

Cone Theorem Step 3

I'm reading the proof of the cone theorem from Koll'ar-Mori and I find step 3 to be very unclear. In particular below is a simple proof of this step which must be wrong somehow. I am wondering where I ...
Daniel's user avatar
  • 5,409
1 vote
0 answers
92 views

About the finite generation of log canonical rings in BCHM

Let $(X,B)$ be a klt pair where $K_X+B$ is $\mathbb{R}$-Cartier. Let $\pi:X\rightarrow U$ be a projective morphism of quasi-projective varieties. Assume either that $B$ is $\pi$-big and $K_X+B$ is $\...
Hobo's user avatar
  • 327
1 vote
1 answer
105 views

Self-intersection of exceptional divisor of blowing-up along a singular point

Let $X$ be an $n$-dimensional projective variety with a simple singularity $p \in X$ which can be resolved by a blow-up $\pi\colon \tilde X \to X$ along $p$. The examples that I am considering are ...
Skadiologist's user avatar
0 votes
1 answer
90 views

Meaning of birational equivalence geometrically

What does birational equivalence signify geometrically? I have been reading AG and my question is why birational equivalence came into the picture? My basic intuition is we may study a space, $X$ not ...
nkh99's user avatar
  • 471
2 votes
1 answer
135 views

Deformation and algebraic equivalence relation

I found the algebraic equivalence relation is closely related to deformation, the most intuitive description that I found is in Griffiths' Topics in Transcendental Algebraic Geometric Chapter 1 which ...
yi li's user avatar
  • 4,828
2 votes
0 answers
22 views

Subadditivity of Kodaira Fiber Dimension

Let $f: X \to C$ be a smooth surjective morphism from a smooth projective complex variety onto a smooth curve, and let $F$ denote a general fiber. I would like to show that $$\kappa(X) \leq \kappa{(F)}...
Daniel's user avatar
  • 5,409
0 votes
0 answers
59 views

Separably rationally connected surface is rational

A variety over a field $k$ is called separably rationally connected if there is a variety Y along with a morphism $u:U=Y \times \mathbb{P}^1 \to X$ such that the map $u \times u : U \times_{Y} U \to X ...
Biman Roy's user avatar
  • 155
0 votes
0 answers
90 views

Very ample + effective = ample?

It is well known that there exist divisors (on a normal projective variety over say the complex numbers) that are big (= is a sum of an ample and an effective divisor) but not ample. However, if we ...
Calculus101's user avatar
3 votes
0 answers
71 views

Bridgeland flops and non-projective flop

In this paper, Bridgeland showed that for a projective 3-fold $X$ with Gorenstein and terminal singularity and a crepant resolution $f:Y \to X$, there exists $g:W \to X$ such that $f$ is the flop of $...
P. Usada's user avatar
  • 400
0 votes
1 answer
243 views

A question in the book of Kollar and Mori

The following comes from [Birational geometry of algebraic varieties] by Kollar and Mori: Here are my questions: 1.(The first red line): Why can they assume $Y$ is smooth at the generic point of $Z_0$...
Hobo's user avatar
  • 327
0 votes
0 answers
54 views

Surface MMP over a curve where the geometric generic fiber is a rational curve

I am looking for an explaination or an reference for the following fact: Let $\pi:X\rightarrow Z$ be a contraction from a smooth surface $X$ to a curve $Z$. Assume that the geometric generic fiber of ...
Hobo's user avatar
  • 327
0 votes
1 answer
51 views

Proper birational morphism with nonreduced exceptional divisor

Let $\pi: Y \to X$ be a proper birational morphism. Let $E$ be the exceptional locus, and $Z = \pi(E)$ the scheme theoretic image (so that it is closed). I want $X,Y,Z,E$ to be smooth. Now take $\pi^{-...
user135743's user avatar
3 votes
1 answer
56 views

Can a birational morphism contract a positive ray?

This question says really screams "I don't know how to compute nef cones and find interesting birational morphisms". If $f:X \to Y$ is a birational morphism between smooth $X,Y$ and an ...
user135743's user avatar
1 vote
1 answer
89 views

Coefficients of members in a base-point free linear system

Let $\mathbb{K}\in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\}$ and let $D$ be a base-point free(or ample if it is necessary) $\mathbb{K}$-divisor on a normal projective variety. I have two questions: When $...
Hobo's user avatar
  • 327
1 vote
1 answer
57 views

If two smooth cubic threefolds are birationally equivalent, are they isomorphic?

Edit: here is a potential proof Let $X,Y$ be two smooth cubic threefolds over an algebraically closed field. Suppose that they are birationally equivalent. Then are they isomorphic? This seemingly ...
TCiur's user avatar
  • 498
1 vote
1 answer
67 views

Example of a quotient of an elliptic curve by a finite group being rational

I am interested in an example of the following situation, over an algebraically closed field o zero characteristic. Let $E$ be an elliptic curve, and $G$ a finite group of automorphisms of $E$ (as an ...
jg1896's user avatar
  • 123
1 vote
1 answer
153 views

Automorphisms of curves and Hurwitz-Riemann formula

Let $k$ be a base field algebraically closed and of zero characteristic. Let $C$ be a smooth projective curve and $G$ a finite group of automorphisms of it. Let $C*$ be a smooth projective curve whose ...
jg1896's user avatar
  • 123
1 vote
0 answers
94 views

Parametrization of algebraic curve using adjoint curves

I am using method described here and here. With curve $\left(x^2+y^2+4 y\right)^2-16 \left(x^2+y^2\right)=0$ it works for me. The curve has singular points $((0,0,1),(-i,1,0),(i,1,0))$ and I used ...
azerbajdzan's user avatar
  • 1,166
3 votes
1 answer
88 views

algebraic fibre space induces algeraically closed extension of function fields

Lazarsfeld said in his book (Positivity in AG 1,example 2.1.12) that if $f:X\rightarrow Y$ is a projective surjective morphism of normal (complex)varieties, and $\mathbb{C}(Y)\subset\mathbb{C}(X)$ the ...
Fan zx's user avatar
  • 51
0 votes
1 answer
63 views

degree, genus, and birational map

I'm confused on the following: Suppose we have a polynomial $f(T)$, e.g. $T^d$, with degree $d>2$, then I consider the curve defined by $y-f(x)$. Then there seems an obvious birational map given by ...
S.Gau at Math's user avatar
0 votes
1 answer
135 views

Is dominant morphism of varities of the same dimension is composition of finite and birational?

I work over field of charateristic zero. I have a morphism $f\colon X\to Y$ of smooth varieties of the same dimension which is proper and dominant. Is it true that $f=g\circ h$ where h is birational ...
Galois group's user avatar
1 vote
1 answer
132 views

History of Algebraic Geometry: Morphisms and Birational Geometry

Good people, I'm trying to get my head around the history of algebraic geometry, and while Dieudonné's tome is a very good source (very often the only source), it can from time to time be very ...
StormyTeacup's user avatar
  • 1,992
4 votes
2 answers
167 views

How to transform genus zero curve to conic section?

I have a curve of genus zero: $$C: 2 x^5-4 x^3 y+x^2 y+2 x y^3+2 x y^2+y^5$$ or in homogenized form: $$C': 2 x^5-4 x^3 y z+x^2 y z^2+2 x y^3 z+2 x y^2 z^2+y^5$$ How to transform it to some conic ...
azerbajdzan's user avatar
  • 1,166
0 votes
1 answer
47 views

Application of ZMT to birational maps in Milne's Algebraic Geometry

I'm studying Algebraic Geometry from this book. At page 187 the author proves that If $\varphi:W\to V$ is a regular birational quasi-finite map between irreducible varieties and $V$ is normal, then $\...
Kandinskij's user avatar
  • 3,644
0 votes
1 answer
56 views

Does $K$-equivalence satisfy transitivity?

There is a series of papers concerning equivalence between $K$-equivalence and $D$-equivalence (DK hypothesis), where two smooth projective varieties $X$ and $Y$ are said to be $K$-equivalent (resp. $...
P. Usada's user avatar
  • 400
2 votes
0 answers
58 views

Converse direction of the Zariski connectness theorem

One has the following Zariski connectness theorem If a proper morphism $\pi: X \rightarrow Y$ of locally Noetherian schemes which has $\mathcal{O}_Y \cong \pi_* \mathcal{O}_X$, then $\pi^{-1}(q)$ is ...
yi li's user avatar
  • 4,828
1 vote
1 answer
94 views

Degree $2$ birational map of $\mathbb{P}^1$

I know that there are many examples of birational degree $2$ projective maps. One example based on the $2$-dimensional Lyness map is $$ \varphi([x,y,z]) = [x y, yz + z^2, x z] $$ with inverse map $$ \...
Somos's user avatar
  • 35.6k
0 votes
1 answer
85 views

Why is the canonical bundle of a hypersurface of degree $d>4$ in $\mathbb {CP}^3$ ample?

Let $F\in \mathbb C[z_0, z_1,z_2, z_3]$ be a homogeneous polynomial of degree $d$ and let $X := \{x \in \mathbb CP^3 \ | \ F([x]) = 0\}$. I have read that if $d>4 $ then the canonical bundle $K_X$ ...
Overflowian's user avatar
  • 5,835
0 votes
0 answers
44 views

How to show that a given mapping is/isn't a birational isomorphism?

I'm currently learning algebraic geometry from Shafarevich's Basic Algebraic Geometry, and a couple of the exercises ask the reader whether or not certain mappings are birational isomorphisms. I know ...
littleman's user avatar
  • 434
0 votes
0 answers
16 views

rational points in singular quartic surface

I came across the following quartic surface in $\mathbb{P}^3$: $S: xyz(x-2y-z)+ww(xy+2xz-yz) = 0$ (It seems that S has singularities at $[x:y:z:w] = [1:0:0:0],[0:1:0:0],[0:0:1:0],[0:0:0:1]$.) I'm ...
aerile's user avatar
  • 1,447
2 votes
1 answer
148 views

If a resolution $f:Y\to X$ satisfies $R^if_*\omega_Y=0$ and $R^if_*\mathcal{O}_Y=0$ for all $i>0$, then do we have $f_*\omega_Y=\omega_X$?

Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic (but I'm mainly interested in positive characteristic). Assume that $X$ has rational singularities,...
imtrying46's user avatar
  • 2,629
0 votes
1 answer
96 views

Existence of a proper birational morphism from a Gorenstein scheme, with trivial higher direct images, implies Cohen-Macaulay? [closed]

Let $R$ be a Noetherian excellent reduced local ring containing a field of characteristic $0$. If there exists a Gorenstein scheme $Y$ and a proper birational map $f: Y \to \text{Spec}(R)$ such that $...
Snake Eyes's user avatar
1 vote
1 answer
57 views

The normalization equals the disjoint union of the normalizations of the irreducible components

I am trying to understand 0CDV from the Stacks Project, whose slogan is the title of this post. The proof invokes two different results and supposedly one can deduce the result from any of them. I am ...
Elías Guisado Villalgordo's user avatar
2 votes
1 answer
92 views

Is a smooth projective variety that is derived equivalent to an abelian variety necessarily an abelian variety?

We say smooth projective varities are derived equivalent if their bounded derived categories of coherent sheaves are equivalent. Thanks to Orlov's work, we know a lot of facts about derived equivalent ...
P. Usada's user avatar
  • 400
4 votes
0 answers
93 views

Fourier-Mukai partners that are birational at every point

Let $X$ and $Y$ be smooth projective irreducible varieties that are Fourier-Mukai partners, i.e., have exact equivalent derived categories $D^b(X) \simeq D^b(Y)$. It is well-known that if a derived ...
P. Usada's user avatar
  • 400
0 votes
0 answers
44 views

Undefined locus for rational map has codimension $\ge 2$, but vanishing locus has codimension 1? [duplicate]

Let $X$ be smooth surface, such that $f:X\dashrightarrow \Bbb{A}^1\subset \Bbb{P}^1$ is a rational map, we have the standard result that the rational map into projective space has undetermined locus ...
yi li's user avatar
  • 4,828

1
2 3 4 5
7