Geodesic
The Fano surface of the Veronese double cone
Izvestiya. Mathematics , Tome 19 (1982) no. 2, pp. 377-443.

Voir la notice de l'article provenant de la source Math-Net.Ru

This article studies the Fano surface $\mathscr F$ of lines on the Veronese double cone $X$ branched in its intersection with a cubic in $P^6$; it is the last variety in the series of Fano 3-folds of index two. The irregularity of the surface $\mathscr F$ is computed, its Abel–Jacobi mapping $\Phi$ into the intermediate Jacobian of the body $X$ is constructed, the Gauss mapping for $\Phi(\mathscr F)$ is studied, and a theorem on uniquely recovering $X$ from $\Phi(\mathscr F)$ is proved. Bibliography: 22 titles. 
@article{IM2_1982_19_2_a9,
     author = {A. S. Tikhomirov},
     title = {The {Fano} surface of the {Veronese} double cone},
     journal = {Izvestiya. Mathematics },
     pages = {377--443},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {1982},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a9/}
}
TY  - JOUR
AU  - A. S. Tikhomirov
TI  - The Fano surface of the Veronese double cone
JO  - Izvestiya. Mathematics 
PY  - 1982
SP  - 377
EP  - 443
VL  - 19
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a9/
LA  - en
ID  - IM2_1982_19_2_a9
ER  - 
%0 Journal Article
%A A. S. Tikhomirov
%T The Fano surface of the Veronese double cone
%J Izvestiya. Mathematics 
%D 1982
%P 377-443
%V 19
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a9/
%G en
%F IM2_1982_19_2_a9
A. S. Tikhomirov. The Fano surface of the Veronese double cone. Izvestiya. Mathematics , Tome 19 (1982) no. 2, pp. 377-443. http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a9/

[1] Iskovskikh V. A., “Antikanonicheskie modeli trekhmernykh algebraicheskikh mnogoobrazii”, Sovremennye problemy matematiki. Itogi nauki i tekhniki, 12, VINITI AN SSSR, M., 1979, 59–157 | MR

[2] Iskovskikh V. A., “Trekhmernye mnogoobraziya Fano, I”, Izv. AN SSSR. Ser. matem., 41:3 (1977), 516–562 | MR | Zbl

[3] Manin Yu. I., Kubicheskie formy: algebra, geometriya, arifmetika, Nauka, M., 1972 | MR

[4] Mamford D., Abelevy mnogoobraziya, Mir, M., 1971

[5] Tikhomirov A. S., “Geometriya poverkhnosti Fano dvoinogo prostranstva $P^3$ s vetvleniem v kvartike”, Izv. AN SSSR. Ser. matem., 44:2 (1980), 415–442 | MR | Zbl

[6] Tikhomirov A. S., “Srednii yakobian dvoinogo prostranstva $P^3$, razvetvlennogo v kvartike”, Izv. AN SSSR. Ser. matem., 44:6 (1980), 1329–1377 | MR | Zbl

[7] Tyurin A. N., “Geometriya osobennostei obschei kvadratichnoi formy”, Izv. AN SSSR. Ser. matem., 44:5 (1980), 1200–1211 | MR | Zbl

[8] Tyurin A. N., “Pyat lektsii o trekhmernykh mnogoobraziyakh”, Uspekhi matem. nauk, 27:5 (1972), 3–50 | Zbl

[9] Tyurin A. N., “Srednii yakobian trekhmernykh mnogoobrazii”, Sovremennye problemy matematiki. Itogi nauki i tekhniki, 12, VINITI AN SSSR, M., 1979, 5–57 | MR

[10] Altman A., Kleiman S., “Introduction to Grothendieck duality theory”, Lecture Notes in Math., 146, Heidelberg, 1970 | MR

[11] Altman A., Kleiman S., “Foundations of theory of Fano schemes”, Compos. Math., 4:1 (1977), 3–47 | MR

[12] Clemens C., Griffiths P., “The intermediate Jacobian of the cubic threefold”, Ann. of Math., 95:2 (1972), 281–356 | DOI | MR | Zbl

[13] Coolidge J., A treatise on algebraic plane curves, Oxford, 1931

[14] Griffiths P., “Periods of integrals on algebraic manifolds, II”, Amer. J. of Math., 90 (1968), 805–865 | DOI | MR | Zbl

[15] Griffiths P., “On the periods of certain rational integrals, I, II”, Annals of Math., 90 (1969), 460–541 | DOI | MR | Zbl

[16] Griffiths P., Harris J., Principles of algebraic geometry, New York, 1977 | MR

[17] Hartshorne R., Algebraic geometry, New York, 1977 | MR

[18] Mori S., “On a generalization of complete intersections”, J. Math. Kyoto Univ., 15:3 (1975), 619–646 | MR | Zbl

[19] Mumford D., “Theta characteristics on an algebraic curve”, Ann. Sci. École Norm. Sup. (4), 4 (1971), 181–192 | MR | Zbl

[20] Room T., The geometry of determinantal loci, Cambridge, 1938 | Zbl

[21] Roth L., Algebraic threefolds, with special regard to problems of rationality, Berlin, Göttingen, Heidelberg, 1955 | MR

[22] Welters G., The Fano surface of lines on a double $P^3$ with 4th order discriminant locus, Preprint, Part I, 1979