# Is there a general geometric characterization for polynomials to be linearly dependent?

Consider $$P$$ the complex projective plane, and fix a line $$L$$ in $$P$$

I had a conjecture, that prof. I. Dolgachev showed me how to prove, that $$3$$ quadratic polynomials depending on a variable $$z \in L$$, say $$p_1$$, $$p_2$$ and $$p_3$$, are linearly dependent over $$\mathbb{C}$$ if and only if:

there exist $$4$$ points $$A$$, $$B$$, $$C$$ and $$D$$ in $$P \setminus L$$, such that the intersections:

$$AB$$ with $$L$$, $$CD$$ with $$L$$

$$AC$$ with $$L$$, $$BD$$ with $$L$$

$$AD$$ with $$L$$, $$BC$$ with $$L$$

are precisely the sets of roots of $$p_1$$, $$p_2$$ and $$p_3$$ respectively. Strictly speaking, one should allow for limiting points too, and so allow for $$A$$, $$B$$, $$C$$ and $$D$$ to degenerate.

Dolgachev's argument is elegant, and uses the fact that $$4$$ points in the plane in general position define a pencil of quadrics, namely the family of quadrics which pass through these points. Such a family contains $$3$$ completely reducible quadrics, which are the $$3$$ pairs of lines mentioned above. Those are the key ingredients. I can provide more details if needed.

My question is, is there a characterization for $$n$$ polynomials of degree $$n-1$$ in one variable $$z \in L$$ (with $$L$$ being a complex projective line) to be linearly dependent over $$\mathbb{C}$$, which is along the same lines as the characterization above for $$n=3$$? For instance, is there a generalization of the criterion above for $$n>3$$, using algebraic geometry?

Edit: for $$n=4$$, the best I could find is the following statement. Consider $$7$$ points $$P_a$$, $$1 \leq a \leq 7$$, in general position in the complex projective plane, and not on $$L$$. Consider a partition of the $$7$$ points into a set $$S$$ of $$5$$ points, and a set $$T$$ of $$2$$ points.

There is a unique quadric passing through all the points in $$S$$, and this quadric intersects $$L$$ at two points, counting multiplicity. On the other hand, there is unique line passing through the two points of $$T$$, which intersects $$L$$ at one point. So each partition of the $$7$$ points into two sets as above gives a divisor of degree $$3$$ on $$L$$ ($$2$$ points coming from $$S$$ and $$1$$ point coming from $$T$$).

If I am not mistaken, if we choose $$4$$ different partitions of the $$7$$ points, then this leads to $$4$$ linearly dependent (over $$\mathbb{C}$$) divisors of degree $$3$$ on $$L$$, and I believe the converse is true.

• When you speak of linear dependence of polynomials depending on $z \in L$, do you mean only that they are linearly independent as functions $L \to \mathbb C$? – LSpice May 25 '19 at 20:07
• I should have made this more precise. By a polynomial of degree $d$ depending on $z \in L$, I mean a holomorphic section of the line bundle $\mathcal{O}(d)$ over $L$, which is a $P^1(\mathbb{C})$. Strictly speaking, knowing the roots only determines the polynomial up to scaling. When I talk about linear dependence, I mean as elements of the vector space $H^0(L, \mathcal{O}(d))$, where each element is up to scaling, thus as elements of the projectivization of that vector space. – Malkoun May 25 '19 at 20:16

For an arbitrary $$n \geq 3$$, consider a collection $$X$$ of $$n(n-1)/2 + 1$$ points in general position in $$P^2(\mathbb{C})$$. Subdivide this collection of points into a set $$S$$ with $$n(n-1)/2 - 1$$ points and a set $$T$$ with the remaining $$2$$ points.
There is a unique planar curve of degree $$n-2$$ passing through all the points in $$S$$, and a unique line passing through the two points in $$T$$. By intersecting with a fixed line $$L$$ in $$P^2(\mathbb{C})$$, this gives in total $$n-2 + 1 = n-1$$ points on $$L$$, counting multiplicity.
Moreover, by counting the dimension of the linear family of degree $$n-1$$ curves passing through the full collection $$X$$, it is clear that if one chooses $$n$$ different partitions of the same full collection $$X$$, then the corresponding $$n$$ divisors of degree $$n-1$$ on $$L$$ are linearly dependent over $$\mathbb{C}$$.
Notice that the family of possible $$X$$s is $$n(n-1) + 2$$ dimensional, while the family of $$n$$ linearly dependent divisors of degree $$n-1$$ on $$L$$ is $$n(n-1) - 1$$ dimensional, so that the family of possible $$X$$s has a larger dimension than the family of $$n$$ linearly dependent divsors of degree $$n-1$$.