# Relations between homogeneous polynomials

Edit: The formulation of my question was incorrect, for several reasons. Here is what I hope to be the correct formulation:

Let $$\mathbb{P}$$ be a projective space, and $$V$$ a general linear subspace of $$H^0(\mathcal{O}_{\mathbb{P}}(d))$$ (that is, a general point in the corresponding Grassmannian). Then for $$p the multiplication map $$H^0(\mathcal{O}_{\mathbb{P}}(p))\otimes V\rightarrow H^0(\mathcal{O}_{\mathbb{P}}(p+d))$$ is of maximal rank, i.e. either injective or surjective.

Is this true? Known? Sasha's answer shows that it is true when $$\dim V \leq \dim \Bbb{P}+1$$.

• I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $\sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y \subsetneq X$? Mar 19, 2019 at 20:57
• Sure. Is that different from what I wrote?
– abx
Mar 19, 2019 at 21:00
• I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written). Mar 19, 2019 at 21:04
• Oh, right. $\ell\leq n$ would be fine for me, though this can be certainly weakened.
– abx
Mar 19, 2019 at 21:43
• $\ell \leq n$ seems far too strong (as in, it makes the question trivial), as then $F_i = x_i^d$ is clearly independent. Mar 19, 2019 at 21:48

I am posting this as an answer since the comment thread is already long. The question is a special case of Fröberg's Conjecture.

MR0813632 (87f:13022)
Fröberg, Ralf(S-STOC)
An inequality for Hilbert series of graded algebras.
Math. Scand. 56 (1985), no. 2, 117–144.
13H15 (13D03 13H10)

This special case is mostly solved by work of Gleb Nenashev.

MR3621254
Nenashev, Gleb(S-STOC)
A note on Fröberg's conjecture for forms of equal degrees.
C. R. Math. Acad. Sci. Paris 355 (2017), no. 3, 272–276.
13D40
https://arxiv.org/pdf/1512.04324.pdf

Theorem 1 proves the maximal rank conjecture for these maps except for a few values of $$p$$ near the "changeover" from injectivity to surjectivity. In particular, Nenashev proves injectivity whenever $$\text{dim} H^0(\mathbb{P},\mathcal{O}_{\mathbb{P}}(p))\otimes V \leq \text{dim} H^0(\mathbb{P},\mathcal{O}_{\mathbb{P}}(p+d)) - \text{dim} H^0(\mathbb{P},\mathcal{O}_{\mathbb{P}}(p))^2,$$ and surjectivity whenever $$\text{dim} H^0(\mathbb{P},\mathcal{O}_{\mathbb{P}}(p))\otimes V \geq \text{dim} H^0(\mathbb{P},\mathcal{O}_{\mathbb{P}}(p+d)) + \text{dim} H^0(\mathbb{P},\mathcal{O}_{\mathbb{P}}(p))^2.$$

• Great, thanks a lot!
– abx
Mar 20, 2019 at 10:35
• You are welcome. Mar 20, 2019 at 10:57

I think this can be controlled as follows. Let $$Z \subset \mathbb{P}^{n-1}$$ be the complete intersection defined by $$F_i$$. Then there is a Koszul resolution $$\dots \to \mathcal{O}(-2d)^{\binom{\ell}{2}} \to \mathcal{O}(-d)^\ell \to \mathcal{O} \to \mathcal{O}_Z \to 0.$$ Twisting it by $$\mathcal{O}(d+p)$$ we obtain $$\dots \to \mathcal{O}(p-d)^{\binom{\ell}{2}} \to \mathcal{O}(p)^\ell \to \mathcal{O}(d+p) \to \mathcal{O}_Z(d+p) \to 0.\tag{*}$$ Your question is equivalent to injectivity of the induced map $$H^0(\mathcal{O}(p)^\ell) \to H^0(\mathcal{O}(d+p)).$$ If $$n \ge \ell$$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?

EDIT (the spectral sequence argument). The hypercohomology spectral sequence of $$(*)$$ has first term $$E_1^{i,j} = H^j\left(\mathcal{O}(d+p+id)^{\binom{\ell}{-i}}\right),\qquad i \le 0$$ and converges to $$E_\infty^k = H^k(\mathcal{O}_Z(d+p))$$. Since a line bundle on a projective space can have only $$H^0$$ or $$H^{n-1}$$, the nonzero terms are only in the rows 0 and $$n-1$$. The leftmost term of the top row is $$E_1^{-\ell,n-1} = H^{n-1}\left(\mathcal{O}(d+p-\ell d)\right)$$ is in the total grading $$-\ell + n - 1 \ge -1$$, hence all differentials from it go to terms of total grading $$\ge 0$$. The same of course is true for the other terms in the top row. On the other hand, the leftmost term in the bottom row is $$E_1^{-1,0} = H^0\left(\mathcal{O}(p)^{\ell}\right)$$ is in the total degree $$-1$$. Thus, no differentials go to this term. Therefore, if the kernel of the differential $$d_1^{-1,0} \colon H^0(\mathcal{O}(p)^\ell) \to H^0(\mathcal{O}(d+p)).$$ is nonzero, it survives in the spectral sequence and gives a contribution to $$E_\infty^{-1} = H^{-1}(\mathcal{O}_Z(d+p)) = 0$$, which is absurd.

• Thanks, but could you explain why the Koszul complex gives that?
– abx
Mar 19, 2019 at 21:03
• Let $E$ the cokernel of $\ldots \rightarrow \mathcal{O}(p-d)^{\binom{l}{2}}$. This truncated comples has length $l-1$. Hence, the dimensions of cohomology groups which play a role in order to compute $H^0(E)$ are less than $l-1$. Mar 19, 2019 at 22:20
• Hence, if you assume that $l \leq n$ and noting that all terms of this truncated complex are acyclic in degree strictly less than $n$, you get that $H^0(E)= 0$ and the injectivity claimed by Sasha. Mar 19, 2019 at 22:22
• @abx: I added an explanation of the spectral sequence argument. Mar 20, 2019 at 6:35
• Thanks very much @Sasha, unfortunately I need a little more. I have changed to a more intrinsic formulation.
– abx
Mar 20, 2019 at 8:14