On varieties in multiple-projective spaces

SummaryIn this paper, mP will denote a projective space of dimension m, and (m,n)P will denote a doubly-projective space of dimension m+n, namely the space of all pairs of points (x|y), where x varies in mP and y in nP. Just so, (m,n,s)P will denote a triply-projective space of dimension m + n + s, and so on.A variety V of dimension d in mP has just one degree g, namely the number of points of intersection of V with d generic linear hyperplanes (ux) = 0, where (ux) means Σuixi. 0n the other hand, a variety V' of dimension d in (m,n)P has several degrees ga,b (a+b=d), defined as follows: ga,b is the number of points of intersection of V' with a hyperplanes (ux) = 0 and b hyperplanes (vy) = 0.Let x0, … xm, y0, … y3 be the homogeneous coordinates of a point in m+n+1P It sometimes happens that the equations of a variety V in m+n+1P are not only homogeneous in all variables x and y together, but even homogeneous in the x's and in the y's. In this case the same set of equations also defines a variety V' in the doubly-projective space (m+n)P. If d is the dimension of V', the dimension of V is for to every point (x|y) of V' corresponds a whole straight line of points (xα, yβ) in V.In some cases it is easier to determine the degrees ga,b of V' than to determine the degree g of V. For this reason, it is desirable to have a rule that enables us to calculate g from the ga,b's. Such a rule will be proved here. It says:The degree g is the sum of all ga,b with a+b=d.In the case of multiply-projective spaces (m,n, …)P the same rule holds: g is the sum of the ga,b, … with a+b+ …=d.Examples of applications of this rule will be given at the end