# Varieties with few trisecant lines

Let $$X\subset\mathbb{P}^N$$ be an irreducible projective variety. Let's denote by $$\mathcal{T}$$ the following property: through a general point $$x\in X$$ there is no line intersecting $$X$$ in at least three points counted with multiplicity (but not contained in $$X$$). I am assuming that $$X$$ is not a hypersurface.

Is there some natural condition on $$X$$ implying that $$\mathcal{T}$$ holds? For instance, if $$X$$ is linearly normal does then $$\mathcal{T}$$ necessarily hold? Is there a classification of the varieties $$X$$ for which $$\mathcal{T}$$ does not hold?

You can have a look at Ingrid Bauer's paper

Bauer, I., The classification of surfaces in $$\mathbb{P}^5$$ having few trisecants, Rend. Semin. Mat., Torino 56, No. 1, 1-20 (1998). ZBL0965.14029.

It turns out that, if a smooth surface $$X \subset \mathbb{P}^5$$ satisfies $$\mathcal{T}$$, then $$\deg X \leq 10$$. Moreover, a fine classification of these surfaces is provided (they belong to eight families).

As a consequence, if $$\deg X \geq 11$$ then $$\mathcal{T}$$ does not hold (and so the answer to your first question is negative).

Assume that $$X$$ is set theoretically defined by quadrics: $$X = Q_1\cap\dots\cap Q_r$$.

If $$L$$ is a line trisecant to $$X$$ then $$L$$ is trisecant to $$Q_i$$ for all $$i$$ and hence $$L\subset Q_i$$ for all $$i$$. So $$L\subset X$$.

Hence $$X$$ does not have any trisecant line besides the lines it containes.

A consequence of Gruson-Peskine $$k$$-secant lemma is the following : if $$2N -3n-1>0$$ then the trisecants of $$X$$ do not fill the ambiant space. In particular, if you know that the secant variety of $$X$$ fills the ambiant space, then you have a numerical condition that guarantees that a general secant is not a trisecant.

• That does now answer to my question. I asked for varieties $X$ such that there is no trisecant line to $X$ through a general point $x\in X$. For instance a rational quartic curve contained in quadric surface in $\mathbb{P}^3$ has infinitely many trisecants and still its secant variety fills the ambient space. Also your numerical condition is satisfied for $N = 3$ and $n = 2$ but any line is trisecant to a surface $X\subset\mathbb{P}^3$ of degree $d\geq 3$. Jul 2 at 18:16
• There is a typo, sorry, it should be $-1$. I will edit. Jul 2 at 18:36