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Let $V_i\subseteq \mathbb{P}^d$ be an irreducible algebraic variety of dimension $d_i$ and degree $n_i$. Bezout's theorem says that if $d_1+d_2=d$ then $$V_1\cap V_2$$ has $$n_1n_2$$ many points (counting multiplicity.)

${\bf Question\ 1:}$ Is it true that if $d_1+d_2+d_3=d$ then $$V_1\cap V_2 \cap V_3$$ has $$n_1n_2n_3$$ many points (counting multiplicity.) ? Have found an old book that affirms this.
Here is an example that refutes it.

${\bf Example:}$ Let the points of $\mathbb{P}^5$ be identified as the coefficients of a projective conic in the plane. Let $P$ be a hyperplane in $\mathbb{P}^5$ consisting of those conics passing through a given point. The statement that a conic is determined by $5$ points I write as $$P^5=1$$ meaning that if I intersect five of these hyperplanes, defined by distinct points of course, but still all denoted by $P$, I get a single point. The intersection of $5$ hyperplanes in $5$ space being a point (in general).

Now let $L$ be the surface of conics tangent to a given line. It is known $$P^4L=2$$ meaning that there are two conics through $4$ given points and tangent to a given line. One sees that this implies that $L$ is a quadratic hypersurface. It follows from Bezout that $$P^3L^2=4$$ and this is ${\it correct}$. It then follows by duality that $$P^2L^3=4$$ $$PL^4=2$$ $$L^5=1$$ and these values are ${\it correct}$.

Jacob Steiner however at one point made the mistake that $$P^2L^3=2^3$$ $$PL^4=2^4$$ $$L^5=2^5$$

But doesnt this follow from the generalized Bezout theorem ?

${\bf Question 2:}$ What is the error is Steiner's supposition ?

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  • $\begingroup$ Fulton Intersection Theory example 9.1.8: the Veronese of double lines (tangent to any conic) account for 31 of the 32 (excess intersection). $\endgroup$ – Jan-Magnus Økland Mar 27 '18 at 19:07
  • $\begingroup$ @Jan-MagnusØkland Thanks for that, so $L^5$ is actually infinite ? $\endgroup$ – Rene Schipperus Mar 27 '18 at 19:16
  • $\begingroup$ @Jan-MagnusØkland If I understand correctly the Veronese surface $V\subseteq L$ for all $L$. And since it is $2$ dimensional one must intersect it with at least, a $P^3$ to get a finite set. So all of the Steiner erroneous values are actually infinite. $\endgroup$ – Rene Schipperus Mar 27 '18 at 19:22
  • $\begingroup$ But there is a way to calculate the equivalency in the example above: $\int_{\rm Veronese} (1+4h)^5\cdot (1-9h+51h^2))=31$, where $h$ is the class of a line in the Veronese$\simeq {\Bbb P}^2.$ $\endgroup$ – Jan-Magnus Økland Mar 27 '18 at 19:27
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    $\begingroup$ The wikipedia article is also fairly detailed: en.wikipedia.org/wiki/Steiner%27s_conic_problem $\endgroup$ – Tabes Bridges Mar 27 '18 at 20:40

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