This question aims to compute ${\rm Vol}(-K_X-tD)$ where $X$ is a $\mathbb{Q}$-factorial Fano variety of dimension $n$ and $D$ is a nonzero effective divisor on $X$. This volume is positive when $0\le t< T(D)$, where $T(D)={\rm sup}\{t>0|-K_X-tD \text{ is big}\}$ is the so-called pseudo-effective threshold.
When $t>0$ is small enough, $-K_X-tD$ is nef and big. Hence the volume is just the self-intersection number $(-K_X-tD)^n$. As $t$ getting large, one may need to use the theory of geography of models introduced by Shokurov. Kaloghiros, Kuronya and Lazic proved the following theorem:
- There exist a sequence of rational numbers $0=T_0<T_1<\cdots <T_m=T(D)$, normal varieties $X_1, \cdots, X_m$ and birational contraction maps $\phi_i: X \dashrightarrow X_i (1\le i\le m)$, such that $\phi_i$ is an ample model of $-K_X-tD$ for $t\in (T_{i-1}, T_i)$.
The sequence $\{(T_i, X_i, \phi_i)\}$ is usually called the ample model sequence of $(X, -K_X; -D)$. Indeed, $X_i$ is given by the big divisor $-K_X-tD$ with $t\in (T_{i-1}, T_i) \cap \mathbb{Q}$: $$X_i={\rm Proj} \bigoplus_{m\ge0} H^0(X, mr(-K_X-tD)), $$ where $r>0$ is an integer such that $r(-K_X-tD)$ is Cartier.
With the theorem in hand, we may compute the volume on each interval: $${\rm Vol}(-K_X-tD)=(-K_{X_i}-tD_i)^n, $$ where $D_i=\phi_{i*}D$ and $ t\in(T_{i-1}, T_i)$.
My question is somehow explicit:
- How can we compute the pseudo-effective threshold and the ample model sequence of an explicit $(X, -K_X, -D)$?
The only example I know is computed by T. Fujita. Let $G=G(2,5)=\mathbb{G}(1,4)$ be the Grassmannian of lines in $\mathbb{P}^4$, $S\subseteq G$ be a plane with $c_2(N_{S/G})=3$. More explicitly, $S$ is the Schubert subvariety $(1,2)\subseteq G$ parametrizing lines in a fixed plane $P\subseteq \mathbb{P}^4$. Blowing up $G$ along $S$ we get a Fano manifold $X$ of dimension $n=6$, and we denote by $E$ the exceptional divisor. Then the ample sequence of $(X, -K_X; -E)$ can be described as: $T_1=5, T_2=T(E)=10$, $X_1=X, D_1=E$, $X_2=\mathbb{P}^6, D_2=H$ where $H$ is a hyperplane of $\mathbb{P}^6$.
In Fujita's computation, it seems that he used the structure of del Pezzo manifolds. I want to know whether we can compute the ample model sequence only using the properties of Grassmannian, for example, Pluker embedding and Schubert calculus. So we may compute other examples with the same method. I would like to know more examples in this topic.
Thank you very much for your answer!
