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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 $\mathcal{O}_X(1)^{\oplus m}$.

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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.

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  • $\begingroup$ 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 $d$? $\endgroup$
    – Jooh
    Aug 12, 2023 at 18:04
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    $\begingroup$ What do you mean by the degree of a non-equidimensional scheme? $\endgroup$
    – Sasha
    Aug 12, 2023 at 18:40

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