Some sequences on projective bundles

Question

Let us consider the projective space bundle $X = \mathbb P(\bigoplus_{i=0}^n \mathcal O_{\mathbb P^m} (b_i))$ on $\mathbb P^m$ with $0< b_0 \leq b_1 \leq...\leq b_n$ and $b = \Sigma_{i=0}^n b_i$. Let $\pi : X \to \mathbb P^m$ be the projection map. let $H= \mathcal O_X(1), F = \pi^{*}(\mathcal O_{\mathbb P^m}(1))$. Then I have the following questions:

$(1)$ It feels intuitively correct but is there a concrete way to see that $\mathcal O_X(F)$ and $\mathcal O_X(H-b_0F)$ are both globally generated ? (and therefore base point free and thus by Bertini we can choose smooth divisors from their corresponding linear systems?)

$(2)$ How to obtain the sequence $0 \to \mathcal O_X(-mF)\to \mathcal O_X^{e_m}(-(m-1)F) \to \cdots \to \mathcal O_X^{e_1} \to \mathcal O_X(F)\to 0$?

where, $e_j:=$ $m+1 \choose j$

It seems that it is a pullback by $\pi$ of a sequence on $\mathbb P^m$, given by $0 \to \mathcal O_{\mathbb P^m}(-m)\to \mathcal O_{\mathbb P^m}^{e_m}(-(m-1)) \to \cdots \to \mathcal O_{\mathbb P^m}^{e_1} \to \mathcal O_{\mathbb P^m}(1)\to 0$? Whats is this sequence called and what are the maps?

$(3)$ How to obtain the exact sequence $0 \to \mathcal O_X(-nH+(b-m-1)F)\to \mathcal O_X^{e_m}(-nH+(b-m)F) \to \cdots \to \mathcal O_X^{e_1}(-nH+(b-1)F) \to \bigoplus_{i=0}^n\mathcal O_X(-(n-1)H+ (b-b_i)F) \to\cdots\to \bigoplus_{|I|=r}\mathcal O_X(-(n-r)H+(b-b_I)F) \to\cdots\to \bigoplus_{i=0}^n\mathcal O_X(b_iF) \to \mathcal O_X(H) \to 0$

where, for $I \subset \{0,...,n\}$, $b_I= \Sigma_{i \in I} b_i$.

This last sequence, I have absolutely no idea how to yield such a sequence. Can someone explain the pattern of terms in the sequence in a specific case?

Any clarification along with a reference (if possible) is welcome.

Answer 1

Sasha's suggestion is correct, that these all come from what is usually known in Algebraic Geometry as the Koszul Complex. This is reviewed, for example, in Appendix B.2 of Lazearfeld's Positivity in Algebraic Geometry I, although the details are mostly left to the reader. The main idea is that there is a global version of the usual Koszul complex, which is associated to a sequence of elements in a ring. (This is covered in your favorite commutative algebra book. For example, in Chapter 17 of Eisenbud's book.)

To summarize, if $s \in H^0(X, E) \setminus \{0\}$ is a nonzero global section of a locally free sheaf $E$ of rank $r$ on a projective variety $X$, with zero set $Z = \{x \in X \;|\;s_x \in \mathfrak{m}_xE_x\}$ there is a naturally defined complex, called the Koszul Complex associated to $s$ $$K^\bullet(s): \;\;0 \longrightarrow \Lambda^r E^\vee \longrightarrow \Lambda^{r - 1} E^\vee \cdots \longrightarrow \Lambda^{1}E^\vee = E^\vee \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}_Z \longrightarrow 0$$ where $E^\vee \to \mathcal{O}_X$ is the map given by evaluation at $s$, and $\mathcal{O}_X \to \mathcal{O}_Z$ is the usual restriction map. This complex is exact if $X$ is Cohen-Macaulay (in particular, smooth).

For example, if $L$ is globally generated by $r$ global sections on $X$, evaluation gives a surjection $(L^\vee)^{\oplus r} \to \mathcal{O}_X$, which is dual to a nonvanishing section $s$ of $L^{\oplus r}$, from which we form a Koszul sequence with terms of the form $$\Lambda^i((L^\vee)^{\oplus r}) \cong ((L^\vee)^{\otimes i})^{\oplus e_i}$$ where $e_i = {r \choose i}$, using the identity here with induction on $r$.

Applying this to $L = \mathcal{O}(F)$ gives an answer to (2), given that $\mathcal{O}(F)$ is generated by the $m + 1$ global sections $\pi^*x_0, \cdots, \pi^*x_m \in H^0(X, \mathcal{O}_X(F))$, which is clear since these sections define a map to projective space. It is also isomorphic to the pullback of the Koszul Complex for $L = \mathcal{O}_{\mathbb{P}^m}(1)$.

As for (3), we have to stitch the Koszul sequence of (2) together with another one.

Recall that if $E$ is a vector bundle on $Y$, with projective bundle $\pi: \mathbb{P}(E) \to Y$, there is always a universal surjection $\pi^*E \to \mathcal{O}_{\mathbb{P}(E)}(1)$ inducing the identity of $\mathbb{P}(E)$. In your case, with $Y = \mathbb{P}^m$ and $E = \mathcal{O}(d_0) \oplus \cdots \oplus \mathcal{O}(d_n)$, this defines a surjection $\bigoplus_{i =0}^n \mathcal{O}(d_iF) \longrightarrow \mathcal{O}(H)$ and hence a surjection $$\bigoplus_{i =0}^n \mathcal{O}(d_iF - H) \longrightarrow \mathcal{O}.$$ Again, this is dual to a nonvanishing section so we can form an exact Koszul sequence where each term is of the form $$\Lambda^j \left( \bigoplus_{i =0}^n \mathcal{O}(d_iF - H) \right) \cong \bigoplus_{|J| = j} \mathcal{O}(d_{J}F - jH)$$ for some $j \in 1, \dots, n + 1$. Again, this isomorphism is given by expanding this exterior product using the aforementioned MSE question. Twisting by $H$ accounts for the first half of the sequence appearing in (3).

To get the other half, take the Koszul sequence from (2) and twist it by $-nH + bF$ to get a complex $$ 0 \longrightarrow \mathcal O_X(-nH+(b-m-1)F)\to \mathcal O_X^{e_m}(-nH+(b-m)F) \longrightarrow \cdots \to \mathcal O_X^{e_1}(-nH+(b-1)F) \longrightarrow \mathcal{O}_X(-nH + bF) \longrightarrow 0.$$

Now, to glue these together to get the sequence of (3), we have to define a map $$\mathcal O_X^{e_1}(-nH+(b-1)F) \longrightarrow \bigoplus_{i = 0}^n \mathcal{O}_X(-(n - 1)H + (b - b_i)F).$$ To do so, we take the composition $$\mathcal O_X^{e_1}(-nH+(b-1)F) \longrightarrow \mathcal{O}_X(-nH + bF) \longrightarrow \bigoplus_{i = 0}^n \mathcal{O}_X(-(n - 1)H + (b - b_i)F)$$ where this right map is the last map is the Koszul complex mentioned before. Since the left map is surjective, the image of the composition is the same as the image of the right map, which is the kernel of the next differential. Similarly, since the right map is injective, the kernel of the composition is the agrees with the image of the preceding differential.

Hence, this composition allows us to form the long exact sequence of (3) by gluing these two Koszul sequences together in this way.