Ideal of the union of two zero loci

Question

Let $X$ be a smooth (complex) projective variety and $\mathcal E$ a globally generated vector bundle on $X$ of rank $< dim(X)$. I would like to know, please, if there is a nice description (exact sequence) of the ideal of the union $Z(\sigma_1)\cup Z(\sigma_2)$ of $2$ distinct zero loci of $\mathcal E$.
(It seems that this ideal should admit a surjection from a subsheaf of $\wedge^2\mathcal E^*$).

Answer 1

Let me consider the case in which $\mathcal{E}$ is a rank 2 bundle.

This doesn't seem necessary, but it should help clarify the situation at least. Second, we need to assume that $Z_1=Z(\sigma_1)$ and $Z_2=Z(\sigma_2)$ meet in the correct dimension (codimension 4 in our case), else I doubt there is much that can be said.

We have a few natural exact sequences.

1. The Koszul complex for $\sigma_1$: $$ 0\rightarrow \det\mathcal{E}^* \xrightarrow{\sigma_1^*}\mathcal{E}^*\rightarrow I_{Z_1}\rightarrow 0. $$
2. The Koszul complex for $\sigma_2$: $$ 0\rightarrow \det\mathcal{E}^* \xrightarrow{\sigma_2^*}\mathcal{E}^*\rightarrow I_{Z_2}\rightarrow 0. $$
3. A short exact sequence relating the ideals of the union and the intersection: $$ 0\rightarrow I_{Z_1 \cup Z_2} \rightarrow I_{Z_1} \oplus I_{Z_2} \rightarrow I_{Z_1\cap Z_2}\rightarrow 0. $$
4. A Koszul complex for $\sigma_1^*\oplus \sigma_2^*$:$$ 0\rightarrow \wedge^4(\mathcal{E}^* \oplus \mathcal{E}^*)\rightarrow \wedge^3(\mathcal{E}^* \oplus \mathcal{E}^*)\rightarrow \wedge^2(\mathcal{E}^* \oplus \mathcal{E}^*)\rightarrow \mathcal{E}^* \oplus \mathcal{E}^*\xrightarrow{\sigma_1^*\oplus \sigma_2^*} I_{Z_1\cap Z_2} \rightarrow 0. $$
5. A short exact sequence coming from 4 (with $K=\operatorname{Ker}(\sigma_1^*\oplus \sigma_2^*)$):$$ 0\rightarrow K \rightarrow \mathcal{E}^* \oplus \mathcal{E}^*\rightarrow I_{Z_1\cap Z_2} \rightarrow 0. $$

Now the sum 1.$\oplus$ 2. maps to sequence 5. Sequence 3. and the snake lemma say that $$ I_{Z_1\cup Z_2} \cong K/(\det\mathcal{E}^*\oplus \det\mathcal{E}^*). $$

Using the Koszul resolution 4, we can resolve $K$. And we can lift the map: $$ \det\mathcal{E}^*\oplus \det\mathcal{E}^* \rightarrow K $$ to a map to the first term in the resolution of $K$: $$ \det\mathcal{E}^*\oplus \det\mathcal{E}^* \rightarrow \wedge^2(\mathcal{E}^* \oplus \mathcal{E}^*). $$ Explicitly this is done by writing $$ \wedge^2(\mathcal{E}^* \oplus \mathcal{E}^*) \cong \det\mathcal{E}^* \oplus (\mathcal{E}^*\otimes \mathcal{E}^*) \oplus \det\mathcal{E}^*. $$ and taking map $$ \pi_1 \oplus 0 \oplus \pi_2 : \det\mathcal{E}^*\oplus \det\mathcal{E}^* \rightarrow \wedge^2(\mathcal{E}^* \oplus \mathcal{E}^*) $$ where $\pi_1$ and $\pi_2$ are projections onto the factors. Altogether we get a 4-term exact sequence: $$ 0\rightarrow \wedge^4(\mathcal{E}^* \oplus \mathcal{E}^*)\rightarrow \wedge^3(\mathcal{E}^* \oplus \mathcal{E}^*)\oplus (\det\mathcal{E}^*\oplus \det\mathcal{E}^*)\rightarrow \wedge^2(\mathcal{E}^* \oplus \mathcal{E}^*)\rightarrow I_{Z_1\cup Z_2} \rightarrow 0. $$