Veronese surfaces in $\mathbb{P}^5$ intersecting finitely

Question

I wondered if it were possible to clarify the basic first lemma in A note on the Intersection of Veronese Surfaces.

Setup: $X_1$ and $X_2$ are two Veronese surfaces in $\mathbb{P}^5$ intersecting along a zero-dimensional scheme $W$.

Statement: Viewing $W$ as a subscheme of $X_1\cong \mathbb{P}^2$, $h^1(I_{W/\mathbb{P}^2}(4))=0$. (So $W$ imposes independent conditions on plane quartics, bounding the length of $W$.)

The proof is just five lines. The argument is to restrict the minimal resolution of $I_{X_2}$ to $X_1$. Using the fact that $I_{X_2}$ is generated by quadrics (with linear relations, linear relations between the relations and so on), we are supposed to conclude something about the resolution of $I_{W/X_1}$.

However, I'm not even sure whether the restriction of the resolution to $X_1$ remains exact or whether that matters.

Answer 1

The restriction of the resolution of $\mathcal{O}_{X_2}$ to $X_1$ computes $\mathrm{Tor}_i(\mathcal{O}_{X_1},\mathcal{O}_{X_2})$. If the intersection $X_1 \cap X_2$ is dimensionally transverse, higher $\mathrm{Tor}_i$ vanish, so the restriction of the resolution is a resolution of $W$.

EDIT. As @DCT observed, if two Veronese surfaces in $\mathbb{P}^5$ intersect, this is not a dimensionally transverse intersection, so one needs to be more careful.

In fact, if the intersection is "almost dimensionally transverse" (in the sense that the dimension of intersection is by one bigger than the expected dimension) then $$ \mathrm{Tor}_{\ge 2}(\mathcal{O}_{X_1},\mathcal{O}_{X_2}) = 0 $$ and $\mathrm{Tor}_1(\mathcal{O}_{X_1},\mathcal{O}_{X_2})$ is an invertible sheaf on the scheme $X_1 \cap X_2$ (this is a local statement, so it can be proved using the Koszul resolution when the schemes $X_1$ and $X_2$ are locally complete intersections and the ambient scheme is Cohen-Macaulay).

In the case of two Veronese surfaces this can be applied as follows. Recall the resolution $$ 0 \to \to \mathcal{O}(-4)^{\oplus 3} \to \mathcal{O}(-3)^{\oplus 8} \to \mathcal{O}(-2)^{\oplus 6} \to \mathcal{O} \to \mathcal{O}_{X_1} \to 0 $$ of a Veronese surface. Restricting it to $X_2$ we obtain the complex $$ 0 \to \to \mathcal{O}_{X_2}(-8)^{\oplus 3} \to \mathcal{O}_{X_2}(-6)^{\oplus 8} \to \mathcal{O}_{X_2}(-4)^{\oplus 6} \to \mathcal{O}_{X_2} \to 0 $$ whose cohomology is $\mathcal{O}_W$ in degrees $0$ and $-1$ (any invertible sheaf on a zero-dimensional scheme is trivial). Twisting by $\mathcal{O}_{X_2}(4)$ we obtain the complex $$ 0 \to \to \mathcal{O}_{X_2}(-4)^{\oplus 3} \to \mathcal{O}_{X_2}(-2)^{\oplus 8} \to \mathcal{O}_{X_2}^{\oplus 6} \to I_W(4) \to 0 $$ whose cohomology is the sheaf $\mathcal{O}_W$ in the term $\mathcal{O}_{X_2}^{\oplus 6}$. Then the hypercohomology spectral sequence gives you $$ 0 \to H^0(X_2,\mathcal{O}_W) \to \mathbb{C}^6 \to H^0(X_2,I_W(4)) \to \mathbb{C}^{9} \to 0 $$ (the last term is $H^2(X_2,\mathcal{O}_{X_2}(-4)^{\oplus 3})$) and $H^{>0}(X_2,I_W(4)) = 0$. In particular, this bounds the length of $W$ by 6.