# Pure-dimensional intersection of smooth varieties

Let $$X\subset \mathbb{P}^n$$ be a complex smooth projective variety of degree $$d$$ and dimension $$m$$, such that $$X$$ is not contained in any projective subspace of $$\mathbb{P}^n$$. Let $$P\subset \mathbb{P}^n$$ be a projective space of dimension $$n-m$$. Then we know that if $$P$$ and $$X$$ intersect properly, $$P\cap X$$ is a zero-dimensional subscheme of length $$d$$.

I'm wondering if some similar properties hold for non-properly intersections. More precisely, I have the following questions.

Question: Suppose that $$X$$ and $$P$$ do not contain each other, and $$\dim P\cap X= 1$$. Assume that $$n$$ and $$m$$ are not too small (for example, take $$n=9, m=6$$).

1. Is the projective scheme $$P\cap X$$ pure-dimensional? I guess the answer is yes, but I don't know how to prove this yet...

2. How to compute the degree of $$P\cap X$$?

3. When is $$P\cap X$$ connected?

I think one of the possible methods is to describe $$P\cap X$$ as the zero locus of a section of $$\mathcal{O}_X(1)^{\oplus m}$$.

No for 1. For instance, let $$X \subset \mathbb{P}^5$$ be the Veronese surface, and let $$P$$ be the intersection of two hyperplanes that correspond to the conics $$C_1 = \{xy = 0\}, \qquad C_2 = \{xz = 0\}$$ in $$X \cong \mathbb{P}^2$$. Then $$X \cap P = C_1 \cap C_2 = \{x = 0\} \cup \{y = z = 0\}$$ is the union of a curve and a point.
• Thanks for your example! How about 2.? Is it possible that $d$ is large but the degree of $P\cap X$ is much smaller than $d$?