# 3-secant lines of a projective curve

Consider a smooth projective curve $$C\subset\mathbb{P}^n$$. Let $$G(1,n)$$ the Grassmannian of lines of $$\mathbb{P}^n$$. The variety $$S_2(C)\subset G(1,n)$$ parametrizing lines that are secant to $$C$$ (i.e., that intersect $$C$$ in at least two points) has dimension two.

Let $$S_3(C)\subset S_2(C)\subset G(1,n)$$ be the variety parametrizing lines that intersect $$C$$ in al least three points.

Question. Is there a formula (perhaps depending on $$n$$, the genus and the degree of $$C$$) for the dimension of $$S_3(C)$$? Under which assumption do we have that $$\dim(S_3(C)) = 0$$?

• The answer depends very much on the embedding. Assuming that $C$ is embedded by a complete linear system $\lvert L\rvert$, $S_3(C)=\varnothing$ is equivalent to $h^0(L(-D_3))=h^0(L)-3$ for all effective divisors $D_3$ of degree 3. This is the case in particular if $\deg(C)\geq 2g+2$; it is also true for the canonical embedding ($L=K$) if $C$ is not hyperelliptic or trigonal.
– abx
Feb 17 at 20:46
• The expected dimension is $4-n$. This should be what you see for a sufficiently general curve $C$ - dimension $1$ if $n=3$, $0$ if $n=4$, and empty for $n>4$. Feb 18 at 0:06

(too long for a comment)

Let's consider the case $$n=3$$.

In $$\mathbb P^3$$, for a curve of genus $$g$$ and degree $$d$$, given a point on $$x\in C$$, lines through $$x$$ are parameterized by $$\mathbb P^2$$.

Lines that intersect another point in $$C$$ are the projection of $$C$$ from $$x$$. This is a curve of genus $$g$$ and degree $$d-1$$ in $$\mathbb P^2$$. (One point on this curve represents a tangent line and not a secant line).

Lines that intersect two other points in $$C$$ are nodes (or other multi-branch singularities) of this curves. The total number of singularities, counted by their contribution to the difference between the arithmetic and geometric genus, is $$(d-2)(d-3)/2 -g$$.

In a sufficiently general situation, these will all be nodes, and so the set of lines (with a marked point) form a degree $$(d-2)(d-3)/2-g$$ cover of $$C$$. I think this is always positive unless $$C$$ is a plane curve (in which case this is trivial), $$g=0$$ and $$d=3$$, or $$g=1$$ and $$d=4$$.

How can we fail to be sufficiently general? Either if the projection map is generically not birational onto its image - that is if a generic line through two points on $$C$$ passes through a third (i.e. if the space of trisecants has dimension $$2$$), or if the general fiber has singularities other than nodes - which happens when a general point on $$C$$ lies on a tangent line (other than its own), or similar degeneracies. Probably one can eliminate these or classify when they happen, but I'm not sure.