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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.

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I know this question has been asked in various forms on this site, but I haven't found the answer to my exact question. For simplicity, let's say I'm blowing up a smooth surface $S$ at a point $p$. ...
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Definition A curve over a field $k$ is a separated scheme $C$ of finite type over $k$ which is integral of dimension 1. Let $C$ be a normal curve over a field $k$. A divisor is an element of the free ...
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I would like to ask the following question. What is the description/example that $f:X \to Y/Z$ is a non-isomorphic proper morphism of normal varieties proper over a normal variety $Z$, such that the ...
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Let $Y$ be a smooth complex Fano variety (i.e. $-K_Y$ is ample), and let $D_1, D_2$ be two Cartier divisors on $Y$. We assume that $D_1, D_2$ are nef, and that $-K_Y-D_1-D_2$ is ample. I want to prove ...
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I am trying to solve Exercise 7.2 in Debarre's Higher Dimensional Algebraic Geometry. The actual exercise is not important for my question; instead, the approach Debarre intends seems to eventually ...
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Let $X$ be a smooth connected proper scheme over base field $k$ and $G=\langle g \rangle$ a finite cyclic group acting on $X$ such that the fixed point locus $D$ under the action by $g$ is a Weil ...
user267839's user avatar
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My question is: Let $f:X\to Y$ be a birational morphism between normal projective varieties such that the exceptional locus of $f$ has codimension $1$ in $X$ ($f$ is a divisorial contraction). Is it ...
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I am studying 6-torsion points on elliptic curves over cubic fields and have obtained a 24-degree algebraic curve. Maple computations confirm that this curve has genus one, despite its high degree. <...
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Let $X$ be a scheme and $L$ a semiample line bundle on it, ie there exist some sections $s_0,..., s_n \in \Gamma(X,L^{\otimes m})$ for some big enough $m \ge 1$ such that $X =\cup X_{s_i}$ (where $X_{...
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Setting Let $X, Y$ be subsets of $\mathbb{A}^n$ (or $\mathbb{P}^n$), and let them be contained by some irreducible closed sets $V_X, V_Y$ such that $X$ is open in $V_X$ and $Y$ is open in $V_Y$. ...
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Suppose $X$ is a projective variety with canonical singularities, and consider the blow-up $Y$ of $X$ along a subvariety $Z$ (in particular I am considering $Z$ to be contained in the singular locus ...
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Let $k$ be a field of characteristic 0. A variety over $k$ for me is a geometrically integral separated $k$-scheme of finite type (over $k$). If $V$ is a $k$-variety, denote by $K(V)$ its field of ...
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I am looking for a reference for the following statement: We work over $\mathbb{C}.$ Let $V_n, V_l \subseteq \mathbb{P}^m$ be non-rational hypersurfaces of degree $n,l \geq m+1.$ Then $V_n$ and $V_l$ ...
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My question is: Is every birational projective morphism $f:Z\to X$ between smooth projective varieties $Z$ and $X$ a blow-up morphism $Bl_YX\to X$ along a subvariety $Y\subset X$? This is the ...
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The following theorem is Hartshorne Chapter V, Proposition 3.6. Let C be an effective divisor on a nonsingular surface X, let P be a point of multiplicity r on C, and let $\pi: \tilde{X}\to X$ be the ...
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Let $X=\mathbb{P}(\mathcal{O}_Y\oplus \mathcal{O}_Y(D))$ be a $\mathbb{P}^1$-bundle over a smooth complex projective variety $Y$, and call the projection map $\pi: X\to Y$. Suppose that there exists a ...
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Let $Y$ be a normal, $\mathbb{Q}$-factorial projective complex variety, let $X=\mathbb{P}(\mathcal{O}_Y(D)\oplus \mathcal{O}_Y(D'))$ be a $\mathbb{P}^1$-bundle over $Y$, and suppose that $D,D'$ are ...
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Let $f: X \to Y$ be a morphism of schemes over $S$. For a points $s \in S$, let $X_s = X \times_S \operatorname{Spec}(k(s))$ and $Y_s$ be the fibers over $s$, and let $f_s$ be the induced morphism $...
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Suppose that $v_0:\mathbb P^1\rightarrow B$ is a free rational curve that admits positive factors. Here, we say $v_0$ is free if in the following decomposition of vector bundles (thanks to the ...
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Let $B^n$ a uniruled manifold (i.e., it admits a free rational curve) and let $\mu:\mathbb P^1\rightarrow B$ be a free rational curve (i.e., $\mu$ is a holomorphic non-constant map). Here, we say $\mu$...
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Let $f:X\rightarrow Y$ be a surjective morphism with connected fibers between normal projective varieties, and let $D\subset Y$ be an $\mathbb{R}$-Cartier divisor. Suppose that $f^*D$ is pseudo-...
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I am studying the example of Atiyah flop, and I have some basic questions which I can't solve or find reference about. We work over $\mathbb{C}$. Let $X$ be the cone over the Segre embedding $\mathbb{...
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Let $\tilde{X}\to X$ be a blowup of a smooth projective variety at smooth locus. Let $Y_1$ and $Y_2$ be two irreducible subvarieties of $X$. Let $\tilde{Y}_1$ and $\tilde{Y}_2$ be the strict ...
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Let $X$ be a smooth projective variety (over the complex numbers), and let $D$ be a prime divisor on $X$. Suppose that $D$ is nef, that is for any irreducible curve $C\subset X$ we have $D\cdot C\geq ...
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Let $X$ be a projective, geometrically connected $k$-surface with a relatively minimal conic bundle structure $X \longrightarrow \mathbb{P}^1_k$. My understanding is that the generic fiber ought to be ...
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I am reading Shavalievich's book Basic Algebraic Geometry. In section 3.3 of chapter $1$, he gives an example to show that a cubic surface containing $2$ skew lines is rational. Here are the details: ...
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Let $f\colon X\rightarrow Y$ be a proper morphism of schemes and $\mathscr{F}$ a coherent $\mathscr{O}_X$-module. Given a point $y\in Y$, we have the following diagram. $$\require{AMScd} \begin{CD} ...
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Let $C$ be a closed convex cone in $\mathbb{R}^n$, and let $l$ be an extremal ray of $C$. Are there only finitely many (exposed)faces containing $l$? (Actually, I care more about the Mori cone.) ...
notime's user avatar
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Let $f_1:X_1 \to \mathbb{P}^1$ and $f_2:X_2 \to \mathbb {P}^1$ be two smooth projective morphisms over $\mathbb{P}^1$. Assume $X_1$ and $X_2$ are smooth varieties. Let $g:X_1\to X_2$ be a morphism ...
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Sorry this is a question due to lack of understanding of blow ups and birationally geometry in general. Consider the hypersurface in $\mathbb{A}^3$ given by: $$X =xy - z^2 = 0,$$ which has a ...
ben huni's user avatar
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Let $X\xrightarrow f Y$ is a birational morphism over positive characteristic, where $X$ is a smooth surface. Assuming we know that $f^*K_Y \cong K_X$, can we conclude that $f^*\Omega_Y\cong\Omega_X$?
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Let $X$ being a smooth variety of dimension $2$ over a field of positive characteristic. Let $X \xrightarrow{f} Y$ be a birational morphism (say blowdown of negative self-intersection curves), and ...
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I am trying to understand resolution of singularities over the complex numbers. What I mean are log resolutions in the sense of Lazarsfeld's book Positivity in Algebraic Geometry I from where I took ...
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I am learning about cyclic covers and here are a few preliminary calculations. I would really appreciate it if someone could kindly check if they are correct, and any comments are appreciated, Let $f:...
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Problem Setup Here are two maps $\mathbb C^3\longrightarrow\mathbb C^3$ $$ \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} \overset{f}{\longmapsto} \begin{...
Chris Wolird's user avatar
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Let $Y\to X$ be a projective birational morphism between normal projective algebraic varieties over $\mathbb{C}$. Further, suppose that $X$ is smooth. Does it then follow that $Y$ is smooth as well? ...
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(cf. "introduction to the Mori program" by K. Matsuki; page 65) Suppose $\alpha:S\to C$ is a smooth morphism from a smooth projective surface to a smooth elliptic curve whose (all) fibers ...
Crisp's user avatar
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If $X$ is a projective scheme with at worst Gorenstein singularities, then the dualising sheaf ${\omega}_X$ is a line bundle and it makes sense to talk about the geometric genus $p_g(X) := \mathrm h^0(...
Skadiologist's user avatar
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Apologies for the vague title. Part of my question is what the technically accurate language is for this function $f$ I'm asking about. The Function I found this map on $\mathbb C^3$ I'd like to ...
Chris Wolird's user avatar
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Let X be a 3 dimensional normal, terminal and Gorenstein variety over non-closed field k of characteristic 0. There is a known result from the 80's by Kawamata stating that with this hypothesis any $\...
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Question Let $F_n = \mathbb{P}(\mathscr{O}_{\mathbb{P}_1}\oplus\mathscr{O}_{\mathbb{P}_1}(n))$ be an Hirzebruch surface and consider its finite covering(or double covering) $f: X \to F_n$. Let $f$ be ...
Tommk's user avatar
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This question seems to claim that a projective birational maps $f: X \to Y$ between varieties $X,Y$ over $\mathbb C$ is a blow-up. What is a reference for that? According to Wikipedia, The "Weak ...
red_trumpet's user avatar
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Let $g: X\to S$ be a proper morphism of analytic space, assume that there exists a big line bundle on $S$ denote it $L_1$ and a $g$-big line bundle $L_2$ defined on $X$, can we construct an (absolute) ...
yi li's user avatar
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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
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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
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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
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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
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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
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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
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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
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