Let X P4 k be a smooth quartic threefold and F the Hilbert scheme of conics lying on X. In their paper, which is divided into three parts, the authors study the geometric properties of F. In the first part the class represented by F in the Hilbert scheme of conics in P4 k is computed, using the methods of B. Tennison [Proc. London Math. Soc. (3) 29 (1974), 714–734; MR0419453 (54 #7474)]. From that, other numerical data of F are derived, namely: the canonical class, the Euler characteristic, the arithmetic genus, and the linear genus. Finally, some results of enumerative type about F are obtained (e.g., through a general point of X there are 972 conics, and the image of the curve F of singular conics, via the Steiner map s: ,!P4 k, has degree 24000). In the second part an analogue of the tangent bundle theorem for the Fano surface of lines on a cubic threefold [cf. C. H. Clemens and P. Griffiths, Ann. of Math. (2) 95 (1972), 281–356; MR0302652 (46 #1796); A. B. Altman and S. L. Kleiman, Compositio Math. 34 (1977), no. 1, 3–47; MR0569043 (58 #27967)] is proven for quartic threefolds. The third part is devoted to the analysis of the conormal bundle of a conic in X. It turns out that the conormal bundle N(C,X) of a conic C on X is locally free of rank 2 and that C is locally a complete intersection on X. From a deep geometric analysis of the conormal bundles N(C,X) the following main result of the paper is derived: (i) If the characteristic of the ground field is different from 2 and 3 then dimF 2. (For char(k) = 0, this result was proven, independently and by other methods, by V. A. Iskovskih [Izv. Akad. Nauk SSSR Ser. Math. 42 (1978), no. 3, 506–549; MR0503430 (80c:14023b)].) (ii) For a general quartic threefold X the Hilbert scheme F is smooth (therefore a smooth irreducible surface if char(k) 6= 2, 3). (iii) H1(C,N(C,X)) = 0 for any conic on a general quartic threefold X. In their introduction the authors point out that the hard part of the numerical study of F, the computation of the geometric genus of the surface F, is still open. Reviewed byWerner Kleinert
On the family of conics lying on a quartic threefold.
COLLINO, Alberto;
1980-01-01
Abstract
Let X P4 k be a smooth quartic threefold and F the Hilbert scheme of conics lying on X. In their paper, which is divided into three parts, the authors study the geometric properties of F. In the first part the class represented by F in the Hilbert scheme of conics in P4 k is computed, using the methods of B. Tennison [Proc. London Math. Soc. (3) 29 (1974), 714–734; MR0419453 (54 #7474)]. From that, other numerical data of F are derived, namely: the canonical class, the Euler characteristic, the arithmetic genus, and the linear genus. Finally, some results of enumerative type about F are obtained (e.g., through a general point of X there are 972 conics, and the image of the curve F of singular conics, via the Steiner map s: ,!P4 k, has degree 24000). In the second part an analogue of the tangent bundle theorem for the Fano surface of lines on a cubic threefold [cf. C. H. Clemens and P. Griffiths, Ann. of Math. (2) 95 (1972), 281–356; MR0302652 (46 #1796); A. B. Altman and S. L. Kleiman, Compositio Math. 34 (1977), no. 1, 3–47; MR0569043 (58 #27967)] is proven for quartic threefolds. The third part is devoted to the analysis of the conormal bundle of a conic in X. It turns out that the conormal bundle N(C,X) of a conic C on X is locally free of rank 2 and that C is locally a complete intersection on X. From a deep geometric analysis of the conormal bundles N(C,X) the following main result of the paper is derived: (i) If the characteristic of the ground field is different from 2 and 3 then dimF 2. (For char(k) = 0, this result was proven, independently and by other methods, by V. A. Iskovskih [Izv. Akad. Nauk SSSR Ser. Math. 42 (1978), no. 3, 506–549; MR0503430 (80c:14023b)].) (ii) For a general quartic threefold X the Hilbert scheme F is smooth (therefore a smooth irreducible surface if char(k) 6= 2, 3). (iii) H1(C,N(C,X)) = 0 for any conic on a general quartic threefold X. In their introduction the authors point out that the hard part of the numerical study of F, the computation of the geometric genus of the surface F, is still open. Reviewed byWerner Kleinert
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