By Leonard Roth

ISBN-10: 3540018891

ISBN-13: 9783540018896

ISBN-10: 3642855318

ISBN-13: 9783642855313

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**Additional info for Algebraic Threefolds: With Special Regard to Problems of Rationality**

**Example text**

The first contains a line, determinable in K, from which it evidently projects into a rational quintic scroll, birational in K, while the second*) projects from a tangent plane into a surface of VERONESE type, since its plane representation is by quartics

Also st passes through the base group (Q(1 5), which does not lie on st*. Through any point P distinct from (Q(IS) and (Q(j 5) there passes a single curve of each of the pencils if>, if>1> if>~. If P is common to st, st*, it is double for the curve Q( of if> which contains it, i. e. it lies on Bs; or it is a contact of this curve with each of the curves of if>1> if>j passing through P; and conversely. In this last case these two curves can coincide with ~, and then P is double for the g(~~l cut by if> on ~; otherwise P lies on 9\; and conversely.

By virtue of these theorems we may speak of a rational curve or surface whenever we are not concerned with the particular extension K' of K in which its birational representation is obtainable. We shall see later that there exist irrational (i. e. non-birational) involutions in any space Sd (d;;::; 3), so that a variety Va (d;;::; 3) which is unirational in K' is not in general birational in any extension oj K'. We illustrate these ideas by examples which will prove useful in the sequel. To begin with, any monoid (primal of order n with a single (n -I)-pIe point) is obviously birational in K; hence, in particular, the VERONESE surface is birational in K; for its general projection on 53 is a monoid (STEINER surface).

### Algebraic Threefolds: With Special Regard to Problems of Rationality by Leonard Roth

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