Codimension two nonsingular subvarieties of quadrics: scrolls and classification in degree $d\leq 10$

Mark Andrea A. de CATALDO
1998 Journal of the Mathematical Society of Japan  
Let $X$ be a codimension two nonsingular subvariety of a nonsingular quadric 2" of dimension $n\geq 5$ . We classify such subvarieties when they are scrolls. We also classify them when the degree $d\leq 10$ . Both results were known when $n=4$. Introduction. The paper [26] completes the classification of scrolls as codimension two subvarieties of projective space $P^{n}$ . Ottaviani's proof consists of three parts. First the sectional genus $g$ is exhibited as a function of the degree $d$ of
more » ... scroll. The degree $d$ is then bounded from above by the use of Castelnuovo-type bounds for $g$ . The final step consists of the construction of varieties with prescribed low invariants which had been accomplished by several authors. In this paper we classify scrolls as codimension two subvarieties of $2^{n}$ ; see Theorem 3.1.2. The analysis is quite similar to the one of [26] with the following three differences. The first one is that there are fourfolds scrolls on $2^{6}$ . The second difficulty is that the method for bounding the degree of scrolls over surfaces on $2^{5}$ of [26] is not sufficient; we go around the problem using lemmata 3.4.2 and 3.4.3. Lastly, once we obtain a maximal list of invariants we must construct all the scrolls in question. This is essentially the problem of constructing varieties of low degree and codimension two on 2 $n$ . We build on the results of [4] and [16] and obtain Theorem 2.1.1, i.e. the complete classification in degree $d\leq 10$ and $n\geq 5$ . This result highlights the role that some special vector bundles on quadrics play in the construction of subvarieties of quadrics. AS a by-pass result of this classification in low degree we are able to construct all scrolls, except for one case: when the degree $d=12$ and the base is a minimal $K3$ surface. We construct an unirational family of these scrolls; see Theorem 3.4.5. We do not know whether or not this is the only one. AS it is explained in the introduction to [10] , the Barth-Larsen Theorem and the double point formulee put severe constraints on varieties embedded in projective space with small codimension. The same is true for any ambient space, so that it is only
doi:10.2969/jmsj/05040879 fatcat:qeuxd7hxbredhnwvcwhsydebge