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I am trying to learn the properties of algebraic varieties defined by bihomogeneous polynomials. Most of the classical references (Hartshorne, Harris, etc) appears to be (at least for me) very brief on this subject. Are there any references that discuss in full details the properties of varieties $X\subset \mathbb{P}^n\times \mathbb{P}^m$ giving a number of examples?

In particular, I am confused by the following apparent paradox as far as the possibility of interpreting one bihomogeneous polynomial of bidegree $(2d,2d)$ in the variables $x_0,...,x_n;y_0,...,y_n$ as a homogeneous polynomial of degree $4d$ in the variables $z_0,...z_{2n+1}$ by means of the identification $z_0\equiv x_0,...,z_{2n+1}\equiv y_n$. To be concrete consider the variety $V_1=V(F_1)\subset \mathbb{P}^5$ defined by the following irreducible polynomial,

$$F = (x_0^{2d}+x_1^{2d}+x_2^{2d})(x_3x_4)^d - (x_3^{2d}+x_4^{2d}+x_5^{2d})(x_0x_1)^d$$,

for any integer $d\ge 1$. The dimension $d_1$ of this hypersurface $\subset \mathbb{P}^5$ is therefore $d_1 = 5-1=4$.

Alternatively, $F$ can be seen as a bihomogeneuos polynomial of bidegree $(2d,2d)$ provided we redefine $x_3\equiv y_0$, $x_4\equiv y_1$, $x_5\equiv y_2$:

$F = (x_0^{2d}+x_1^{2d}+x_2^{2d})(y_0y_1)^d - (y_0^{2d}+y_1^{2d}+y_2^{2d})(x_0x_1)^d$

Now, we have the variety $V_2=V(F)\subset \mathbb{P}^2\times\mathbb{P}^2$. By applying the Segre embedding $z_{ij}=x_i y_j$ from $\mathbb{P}^2\times\mathbb{P}^2$ to $\mathbb{P}^8$ we can conclude that the dimension $d_2$ of $V_2$ is $d_2=4-1=3$. This is because should be at least one less the dimension of the Segre embedding $\mathbb{P}^2\times\mathbb{P}^2 \rightarrow \mathbb{P}^8$ (which is four).

Question 1: Different ``embedding'' of the same polynomial can lead us to varieties with distinct dimension? This appears to be a paradox. Where I am making a mistake?

Another point is the following. It turns out that a possible solution for the polynomial $F$ is to separate the variables $x$ from $y$, i.e., we substitute the polynomial $F$ by two polynomials $G_1$ and $G_2$,

$G_1=x_0^{2d}+x_1^{2d}+x_2^{2d}-cx_0^dx_1^d,~~G_2=y_0^{2d}+y_1^{2d}+y_2^{2d}-cy_0^dy_1^d$

where $c$ is a constant. The variety $V(F)$ is now given by $V_3 = V(F) = V(G_1)\cup V(G_2)$. The dimension $d_3$ of $V_3 $ is clearly $d_3 = 1+1=2$ since we have two independent curves $\subset \mathbb{P}^2$.

Question 2: Is there any special terminology in algebraic geometry that describe subvarieties obtained in such way (separation of variables)? Any developed theory which assures us conditions for the existence of such particular solutions in bihomogeneuos varieties? Should them be called diagonal solutions or this is not the appropriate name?

Regards

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Whilst I am sure everyone on this site can sympathise with you in remembering the difficulty of trying to learn algebraic geometry for the first time, I don't think this question is research level and hence not appropiate for this site. You might fare better on some of the sites listed in the FAQ, such as stack exchange. –  Daniel Loughran Jan 14 '12 at 0:28
    
math.stackexchange.com is the URL of the site Daniel is referring to. –  Mariano Suárez-Alvarez Jan 14 '12 at 1:19
    
Eisenbud's "Commutative algebra with a view..." treats these topics, related to elimination theory, quite a bit. –  Dima Pasechnik Jan 14 '12 at 4:05
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1 Answer

pepper,

I'm afraid that there is a more serious problem here than just asking on the wrong forum.

You are making fundamental mistakes that don't even belong to algebraic geometry. You seem to have problems grasping basic mathematical concepts.

1) If you define $F$ as $(x_0^{2d}+x_1^{2d}+x_2^{2d})(x_3x_4)^d - (x_3^{2d}+x_4^{2d}+x_5^{2d})(x_0x_1)^d$, then it is not equal to $(x_0^{2d}+x_1^{2d}+x_2^{2d})(y_0y_1)^d - (y_0^{2d}+y_1^{2d}+y_2^{2d})(x_0x_1)^d$. They live in different worlds (rings) and there is no isomorphism between these two rings, let alone one that would take one polynomial to the other.

2) It does not make sense to talk about different "embeddings" of the same polynomial. You could talk about different embeddings of the zero set of a polynomial.

3) $V(F)\neq V(G_1)\cup V(G_2)$ (for either definition of $F$). Are you saying that $g_1+g_2=0$ iff $g_1=0$ or $g_2=0$? Are you mixing up addition and multiplication?

4) As far as the theory of bihomogenous varieties goes, in my opinion, the reason for the lack of references is that the general theory covers this case pretty well. Using the Segre embedding you can consider them "traditional" varieties. The fact that there are two sets of coordinates just means that the Picard group is not rank one, or if you want that it is $\mathbb Z\times \mathbb Z$. Or even further that there are two maps on your variety that are transverse to each other. I think you can get pretty much everything you want out of these simple assumptions. At the same time, I think there is a short section on these in Fulton's book on algebraic curves and I am sure there is more in the classical literature. Try older books.

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