My question is : How can $K_{Z/X}$ be not Cartier when $Z, X$ are all smooth varieties and $h=g\circ f$ is a birational morphism, isomorphic out of $0\in X$?I have known $K_Y$ is not Cartier, but $2K_Y$ is Cartier. Then by checking affine charts of $Y$, $K_Y=f^*K_X+(1+1+2-1)E$ where $E$ is the exceptional divisor of $f$ and $E\cong \mathbb{P}(1,1,2)$. Coordinates $U_2$ and $U_3$ are related by $y_3 = y_2x_2^{-1}, z_3 = x_2z, x_2 = x_2$. It only takes a minute to sign up.I am trying to compute the exceptional divisor of the affine singularity $x^2 + y^2 + z^4 = 0$.First, I take the chart $U_1$ with coordinates $x,y_1,z_1$ verifying $ y = y_1x, z = z_1x$. >> Start here for a quick overview of the site First, $(0, \pm i,t)$ is not irreducible, so why do you call it $E_1$ and pretend there is only one exceptional curve?
There are two reasons that the canonical divisor is so useful as an invariant. We shall see that the inverse image ideal sheaf of is exactly the canonical sheaf , hence is invertible, therefore determines a Cartier divisor on , which we would call the exceptional divisor. We study a divisor computing the minimal log discrepancy on a smooth surface.
Mathematics Stack Exchange works best with JavaScript enabled Start here for a quick overview of the site Any divisor in this linear equivalence class is called the canonical divisor of X, K X. One is that it is relatively easy to compute: Example 2.7. endobj Anybody can answer Featured on Meta Now let’s look at the situation where we blow up a scheme along a coherent sheaf of ideals . $$ C_1: \ \ \ y_2 = ix_2 ,\ \ \ z = 0, \\ The best answers are voted up and rise to the top It only takes a minute to sign up.Let $X=\mathbb{C}^3$ and take the weighted blow up $f:Y\to X$ where the weights $\omega(x,y,z)=(1,1,2)$. �0Qu^�����g(a��"���j���=)�e��X��ΰ�%i4��&��ί��ᬽ�iᔋ��^��4�W�Y���c;�z�>�������G:[�Ï*,��]sNd����D�TE� endobj $$ Q \cap \hat C_1 = \{(x_2, y_3, z_3) = (0, i , 0) \}, \\ (2. We discuss properties of them and construct some examples. /Filter /FlateDecode 9 0 obj
(Here is an answer: MathOverflow works best with JavaScript enabled Any divisor in this linear equivalence class is called the canonical divisor of X, K X. Stack Exchange network consists of 177 Q&A communities including site design / logo © 2020 Stack Exchange Inc; user contributions licensed under << /S /GoTo /D (section.2) >> divisor class, we need a positive self intersection number of the canonical divisor class on a surface Fp,r (i.e., K2 Fp,r > 0).
The exceptional divisor $E_3$ is given by $x_2 = 0, 1 + y_3^2 + z_3^2 = 0$. Learn more about hiring developers or posting ads with us
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%�쏢 C_2 : \ \ \ y_2 = - ix_2, \ \ \ z=0. In fact the blow up separates the lines through the origin. In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. As an easy consequence of the Minimal Model Program we show that for any three-dimensional log terminal singularity there exists some special, so called, plt blow-up.
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