On the existence of Enriques-Fano threefolds of index greater than one
Authors:
Luis Giraldo, Angelo Felice Lopez and Roberto Muñoz
Journal:
J. Algebraic Geom. 13 (2004), 143-166
DOI:
https://doi.org/10.1090/S1056-3911-03-00342-4
Published electronically:
September 17, 2003
MathSciNet review:
2008718
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References |
Additional Information
Abstract: Let $X \subset \mathbb {P}^{N}$ be an irreducible threefold having a hyperplane section $Y$ that is a smooth Enriques surface and such that $X$ is not a cone over $Y$. In 1938 Fano claimed a classification of such threefolds; however, due to gaps in his proof, the problem still remains open. In this article we solve the case when $Y$ is the $r$-th Veronese embedding, for $r \geq 2$, of another Enriques surface, by proving that there are no such $X$. The latter is achieved, among other things, by a careful study of trisecant lines to Enriques surfaces. As another consequence we get precise information on the ideal of an Enriques surface. In a previous paper we had proved that any smooth linearly normal Enriques surface has homogeneous ideal generated by quadrics and cubics. Here we are able to specify when the quadrics are enough, at least scheme-theoretically.
[Al]Al Alexeev, V.: General elephants of ${\mathbb Q}$-Fano 3-folds. Compositio Math. 91, (1994) 91-116.
[ArSe]ArSe Arbarello, E., Sernesi, E.: Petri’s approach to the study of the ideal associated to a special divisor. Invent. Math. 49, (1978) 99-119.
[ArSo]ArSo Arrondo, E., Sols, I.: On congruences of lines in the projective space. Mém. Soc. Math. France 50, (1992).
[Ba]Ba Bayle, L.: Classification des variétés complexes projectives de dimension trois dont une section hyperplane générale est une surface d’Enriques. J. Reine Angew. Math. 449, (1994) 9-63.
[Bd]Bd Badescu, L.: Polarized varieties with no deformations of negative weights. In: Geometry of complex projective varieties (Cetraro, 1990), Sem. Conf. 9, Mediterranean, Rende (Italy), 1993, 9-33.
[BEL]BEL Bertram, A., Ein, L., Lazarsfeld, R.: Surjectivity of Gaussian maps for line bundles of large degree on curves. In: Algebraic Geometry (Chicago, IL, 1989), Lecture Notes in Math. 1479, Springer, Berlin-New York: 1991, 15-25.
[Bo]Bo Bogomolov, F.: Holomorphic tensors and vector bundles on projective varieties. Izv. Akad. Nauk SSSR Ser. Mat. 42, (1978) 1227-1287, 1439; English transl., Math. USSR-Izv. 13, (1979) 499–555.
[BPV]BPV Barth, W., Peters, C., van de Ven, A.: Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Springer-Verlag, Berlin-New York, 1984.
[CD1]CD1 Cossec, F., Dolgachev, I.: Enriques surfaces I. Progress in Mathematics 76, Birkhäuser Boston, MA, 1989.
[CD2]CD2 Cossec, F., Dolgachev, I.: Enriques surfaces II. To appear.
[Ch1]Ch1 Cheltsov, I.A.: Three-dimensional algebraic varieties that have a divisor with a numerically trivial canonical class. Russian Math. Surveys 51, (1996) 140-141.
[Ch2]Ch2 Cheltsov, I.A.: On the rationality of non-Gorenstein ${\mathbb Q}$-Fano $3$-folds with an integer Fano index. In: Birational algebraic geometry (Baltimore, MD, 1996), Contemp. Math. 207, Amer. Math. Soc., Providence, RI, 1997, 43-50.
[Ch3]Ch3 Cheltsov, I.A.: Boundedness of Fano 3-folds of integer index. Math. Notes 66, (1999) 360-365.
[CiVe]CiVe Ciliberto, C., Verra, A.: On the surjectivity of the Gaussian map for Prym-canonical line bundles on a general curve. In: Geometry of complex projective varieties (Cetraro, 1990), Sem. Conf. 9, Mediterranean, Rende (Italy), 1993, 117-141.
[CLM1]CLM1 Ciliberto, C., Lopez, A.F., Miranda, R.: Projective degenerations of K3 surfaces, Gaussian maps and Fano threefolds. Invent. Math. 114, (1993) 641-667.
[CLM2]CLM2 Ciliberto, C., Lopez, A.F., Miranda, R.: Classification of varieties with canonical curve section via Gaussian maps on canonical curves. Amer. J. Math. 120, (1998) 1-21.
[Co]Co Cossec, F.R.: On the Picard group of Enriques surfaces. Math. Ann. 271, (1985) 577-600.
[Co1]Co1 Conte, A.: Two examples of algebraic threefolds whose hyperplane sections are Enriques surfaces. In: Algebraic geometry—open problems (Ravello, 1982), Lecture Notes in Math. 997, Springer, Berlin-New York, 1983, 124-130.
[Co2]Co2 Conte, A.: On the nature and the classification of Fano-Enriques threefolds. In: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc. 9, Bar-Ilan Univ., Ramat Gan, 1996, 159-163.
[CoMa]CoMa Coppens, M., Martens, G.: Secant spaces and Clifford’s theorem. Compositio Math. 78, (1991) 193-212.
[CoMu]CoMu Conte, A., Murre, J.P.: Algebraic varieties of dimension three whose hyperplane sections are Enriques surfaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12, (1985) 43-80.
[CoVe]CoVe Conte, A., Verra, A.: Reye constructions for nodal Enriques surfaces. Trans. Amer. Math. Soc. 336, (1993) 79-100.
[DR]DR Dolgachev, I., Reider, I.: On rank $2$ vector bundles with $c_{1}^2 = 10$ and $c_{2} = 3$ on Enriques surfaces. In: Algebraic geometry (Chicago, IL, 1989), Lecture Notes in Math. 1479, Springer, Berlin, 1991, 39-49.
[E]E Ein, L.: The irreducibility of the Hilbert scheme of smooth space curves. Proceedings of Symposia in Pure Math. 46, (1987) 83-87.
[ELMS]ELMS Eisenbud, D., Lange, H., Martens, G., Schreyer, F.O.: The Clifford dimension of a projective curve. Compositio Math. 72, (1989) 173-204.
[Fa1]Fa1 Fano, G.: Sopra alcune varietà algebriche a tre dimensioni aventi tutti i generi nulli. Atti Accad. Torino 43, (1908) 973-984.
[Fa2]Fa2 Fano, G.: Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulli. Atti Congr. Internaz. Bologna IV, (1929) 115-121.
[Fa3]Fa3 Fano, G.: Su alcune varietà algebriche a tre dimensioni a curve sezioni canoniche. Scritti Mat. offerti a L. Berzolari. Ist. Mat. R. Univ. Pavia, Pavia, (1936) 329-349.
[Fa4]Fa4 Fano, G.: Sulle varietà algebriche a tre dimensioni a curve sezioni canoniche. Mem. Accad. d’Italia VIII, (1937) 23-64.
[Fa5]Fa5 Fano, G.: Nuove ricerche sulle varietà algebriche a tre dimensioni a curve-sezioni canoniche. Pont. Acad. Sci. Comment. 11, (1947) 635-720.
[Fa6]Fa6 Fano, G.: Sulle varietà algebriche a tre dimensioni le cui sezioni iperpiane sono superficie di genere zero e bigenere uno. Memorie Soc. dei XL 24, (1938) 41-66.
[Fu]Fu Fujita, T.: Rational retractions onto ample divisors. Sci. Papers College Arts Sci. Univ. Tokyo 33, (1983) 33-39.
[GH]GH Griffiths, P., Harris, J.: Residues and zero-cycles on algebraic varieties. Ann. of Math. 108, (1978) 461-505.
[GLM]GLM Giraldo, L., Lopez, A.F., Muñoz, R.: On the projective normality of Enriques surfaces (with an appendix by Lopez, A.F. and Verra, A.). Math. Ann. 324, (2002) 135–158.
[Go1]Go1 Godeaux, L.: Sur les variétés algébriques à trois dimensions dont les sections hyperplanes sont des surfaces de genre zéro et de bigenre un. Bull. Acad. Belgique Cl. Sci. 14, (1933) 134-140.
[Go2]Go2 Godeaux, L.: Une variété algébrique à trois dimensions dont les sections hyperplanes sont des surfaces de bigenre un. Bull. Soc. Roy. Sci. Liége 31, (1962) 751-756.
[Go3]Go3 Godeaux, L.: Sur les variétés algébriques à trois dimensions dont les sections hyperplanes sont des surfaces de bigenre un. Acad. Roy. Belg. Bull. Cl. Sci. 48, (1962) 1251-1257.
[Go4]Go4 Godeaux, L.: Une variété algébrique à trois dimensions à sections hyperplanes de bigenre un. Bull. Soc. Roy. Sci. Liége 39, (1970) 439-441.
[Gr]Gr Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differential Geom. 19, (1984) 125-171.
[H]H Hulek, K.: Projective geometry of elliptic curves. Astérisque 137 (1986).
[I1]I1 Iskovskih, V.A.: Fano 3-folds I. Izv. Akad. Nauk SSSR Ser. Mat. 41, (1977) 516-562, 717; English transl., Math. USSR-Izv. 11, (1977) 485–527.
[I2]I2 Iskovskih, V.A.: Fano 3-folds II. Izv. Akad. Nauk SSSR Ser. Mat. 42, (1978) 506-549; English transl., Math. USSR-Izv. 12, (1978) 469–506.
[I3]I3 Iskovskih, V.A.: Anticanonical models of three-dimensional algebraic varieties. Current problems in mathematics 12, VINITI, Moscow, 1979, 59-157, 239; English transl., J. Soviet Math. 13, (1980) 745–814.
[I4]I4 Iskovskih, V.A.: Birational automorphisms of three-dimensional algebraic varieties. Current problems in mathematics 12, VINITI, Moscow, 1979, 159-236, 239; English transl., J. Soviet Math. 13, (1980) 815–868.
[K]K Kleppe, J.O.: Liaison of families of subschemes in $\mathbb {P}^n$. In: Algebraic curves and projective geometry (Trento, 1988), Lecture Notes in Math. 1389, Springer, Berlin, 1989, 128-173.
[La]La Lazarsfeld, R.: A sampling of vector bundle techniques in the study of linear series. Lectures on Riemann surfaces (Trieste, 1987), World Sci. Publishing, Teaneck, NJ, 1989, 500-559.
[LS]LS Lange, H., Sernesi, E.: Quadrics containing a Prym-canonical curve. J. Algebraic Geom. 5, (1996) 387-399.
[Lv]Lv L’vovsky, S.: Extensions of projective varieties and deformations. I. Michigan Math. J. 39, (1992) 41-51.
[Ma]Ma Martens, G.: Über den Clifford-Index algebraischer Kurven. J. Reine Angew. Math. 336, (1982) 83-90.
[Mf]Mf Mumford, D.: Varieties defined by quadratic equations. In: Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, (1970) 29-100.
[MM]MM Mori, S., Mukai, S.: Classification of Fano $3$-folds with $B_ {2}\geq 2$. Manuscripta Math. 36, (1981/82) 147-162.
[Mu]Mu Murre, J.P.: Classification of Fano threefolds according to Fano and Iskovskih. In: Algebraic threefolds (Varenna, 1981), Lecture Notes in Math. 947, Springer, Berlin, 1982, 35-92.
[P]P Prokhorov, Y.G.: On three-dimensional varieties with hyperplane sections-Enriques surfaces. Mat. Sb. 186, (1995) 113-124; English transl., Sb. Math. 186, (1995) 1341–1352.
[R]R Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. of Math. 127, (1988) 309-316.
[Sa1]Sa1 Sano, T.: On classification of non-Gorenstein ${\mathbb Q}$-Fano $3$-folds of Fano index $1$. J. Math. Soc. Japan 47, (1995) 369-380.
[Sa2]Sa2 Sano, T.: Classification of non-Gorenstein ${\mathbb Q}$-Fano $d$-folds of Fano index greater than $d-2$. Nagoya Math. J. 142, (1996) 133-143.
[Sc]Sc Schreyer, F.O.: Syzygies of canonical curves and special linear series. Math. Ann. 275, (1986) 105-137.
[Sw]Sw Schwartau, P.W.: Liaison addition and monomial ideals. Ph.D. Thesis, Brandeis University, Waltham, MA, (1982).
[W1]W1 Wahl, J.: Introduction to Gaussian maps on an algebraic curve. In: Complex Projective Geometry, Trieste-Bergen 1989, London Math. Soc. Lecture Notes Ser. 179, Cambridge Univ. Press, Cambridge 1992, 304-323.
[W2]W2 Wahl, J.: Gaussian maps on algebraic curves. J. Differential Geom. 32, (1990) 77-98.
[Z]Z Zak, F.L.: Some properties of dual varieties and their application in projective geometry. In: Algebraic Geometry (Chicago, IL, 1989), Lecture Notes in Math. 1479, Springer, Berlin 1991, 273-280.
[Al]Al Alexeev, V.: General elephants of ${\mathbb Q}$-Fano 3-folds. Compositio Math. 91, (1994) 91-116.
[ArSe]ArSe Arbarello, E., Sernesi, E.: Petri’s approach to the study of the ideal associated to a special divisor. Invent. Math. 49, (1978) 99-119.
[ArSo]ArSo Arrondo, E., Sols, I.: On congruences of lines in the projective space. Mém. Soc. Math. France 50, (1992).
[Ba]Ba Bayle, L.: Classification des variétés complexes projectives de dimension trois dont une section hyperplane générale est une surface d’Enriques. J. Reine Angew. Math. 449, (1994) 9-63.
[Bd]Bd Badescu, L.: Polarized varieties with no deformations of negative weights. In: Geometry of complex projective varieties (Cetraro, 1990), Sem. Conf. 9, Mediterranean, Rende (Italy), 1993, 9-33.
[BEL]BEL Bertram, A., Ein, L., Lazarsfeld, R.: Surjectivity of Gaussian maps for line bundles of large degree on curves. In: Algebraic Geometry (Chicago, IL, 1989), Lecture Notes in Math. 1479, Springer, Berlin-New York: 1991, 15-25.
[Bo]Bo Bogomolov, F.: Holomorphic tensors and vector bundles on projective varieties. Izv. Akad. Nauk SSSR Ser. Mat. 42, (1978) 1227-1287, 1439; English transl., Math. USSR-Izv. 13, (1979) 499–555.
[BPV]BPV Barth, W., Peters, C., van de Ven, A.: Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Springer-Verlag, Berlin-New York, 1984.
[CD1]CD1 Cossec, F., Dolgachev, I.: Enriques surfaces I. Progress in Mathematics 76, Birkhäuser Boston, MA, 1989.
[CD2]CD2 Cossec, F., Dolgachev, I.: Enriques surfaces II. To appear.
[Ch1]Ch1 Cheltsov, I.A.: Three-dimensional algebraic varieties that have a divisor with a numerically trivial canonical class. Russian Math. Surveys 51, (1996) 140-141.
[Ch2]Ch2 Cheltsov, I.A.: On the rationality of non-Gorenstein ${\mathbb Q}$-Fano $3$-folds with an integer Fano index. In: Birational algebraic geometry (Baltimore, MD, 1996), Contemp. Math. 207, Amer. Math. Soc., Providence, RI, 1997, 43-50.
[Ch3]Ch3 Cheltsov, I.A.: Boundedness of Fano 3-folds of integer index. Math. Notes 66, (1999) 360-365.
[CiVe]CiVe Ciliberto, C., Verra, A.: On the surjectivity of the Gaussian map for Prym-canonical line bundles on a general curve. In: Geometry of complex projective varieties (Cetraro, 1990), Sem. Conf. 9, Mediterranean, Rende (Italy), 1993, 117-141.
[CLM1]CLM1 Ciliberto, C., Lopez, A.F., Miranda, R.: Projective degenerations of K3 surfaces, Gaussian maps and Fano threefolds. Invent. Math. 114, (1993) 641-667.
[CLM2]CLM2 Ciliberto, C., Lopez, A.F., Miranda, R.: Classification of varieties with canonical curve section via Gaussian maps on canonical curves. Amer. J. Math. 120, (1998) 1-21.
[Co]Co Cossec, F.R.: On the Picard group of Enriques surfaces. Math. Ann. 271, (1985) 577-600.
[Co1]Co1 Conte, A.: Two examples of algebraic threefolds whose hyperplane sections are Enriques surfaces. In: Algebraic geometry—open problems (Ravello, 1982), Lecture Notes in Math. 997, Springer, Berlin-New York, 1983, 124-130.
[Co2]Co2 Conte, A.: On the nature and the classification of Fano-Enriques threefolds. In: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc. 9, Bar-Ilan Univ., Ramat Gan, 1996, 159-163.
[CoMa]CoMa Coppens, M., Martens, G.: Secant spaces and Clifford’s theorem. Compositio Math. 78, (1991) 193-212.
[CoMu]CoMu Conte, A., Murre, J.P.: Algebraic varieties of dimension three whose hyperplane sections are Enriques surfaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12, (1985) 43-80.
[CoVe]CoVe Conte, A., Verra, A.: Reye constructions for nodal Enriques surfaces. Trans. Amer. Math. Soc. 336, (1993) 79-100.
[DR]DR Dolgachev, I., Reider, I.: On rank $2$ vector bundles with $c_{1}^2 = 10$ and $c_{2} = 3$ on Enriques surfaces. In: Algebraic geometry (Chicago, IL, 1989), Lecture Notes in Math. 1479, Springer, Berlin, 1991, 39-49.
[E]E Ein, L.: The irreducibility of the Hilbert scheme of smooth space curves. Proceedings of Symposia in Pure Math. 46, (1987) 83-87.
[ELMS]ELMS Eisenbud, D., Lange, H., Martens, G., Schreyer, F.O.: The Clifford dimension of a projective curve. Compositio Math. 72, (1989) 173-204.
[Fa1]Fa1 Fano, G.: Sopra alcune varietà algebriche a tre dimensioni aventi tutti i generi nulli. Atti Accad. Torino 43, (1908) 973-984.
[Fa2]Fa2 Fano, G.: Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulli. Atti Congr. Internaz. Bologna IV, (1929) 115-121.
[Fa3]Fa3 Fano, G.: Su alcune varietà algebriche a tre dimensioni a curve sezioni canoniche. Scritti Mat. offerti a L. Berzolari. Ist. Mat. R. Univ. Pavia, Pavia, (1936) 329-349.
[Fa4]Fa4 Fano, G.: Sulle varietà algebriche a tre dimensioni a curve sezioni canoniche. Mem. Accad. d’Italia VIII, (1937) 23-64.
[Fa5]Fa5 Fano, G.: Nuove ricerche sulle varietà algebriche a tre dimensioni a curve-sezioni canoniche. Pont. Acad. Sci. Comment. 11, (1947) 635-720.
[Fa6]Fa6 Fano, G.: Sulle varietà algebriche a tre dimensioni le cui sezioni iperpiane sono superficie di genere zero e bigenere uno. Memorie Soc. dei XL 24, (1938) 41-66.
[Fu]Fu Fujita, T.: Rational retractions onto ample divisors. Sci. Papers College Arts Sci. Univ. Tokyo 33, (1983) 33-39.
[GH]GH Griffiths, P., Harris, J.: Residues and zero-cycles on algebraic varieties. Ann. of Math. 108, (1978) 461-505.
[GLM]GLM Giraldo, L., Lopez, A.F., Muñoz, R.: On the projective normality of Enriques surfaces (with an appendix by Lopez, A.F. and Verra, A.). Math. Ann. 324, (2002) 135–158.
[Go1]Go1 Godeaux, L.: Sur les variétés algébriques à trois dimensions dont les sections hyperplanes sont des surfaces de genre zéro et de bigenre un. Bull. Acad. Belgique Cl. Sci. 14, (1933) 134-140.
[Go2]Go2 Godeaux, L.: Une variété algébrique à trois dimensions dont les sections hyperplanes sont des surfaces de bigenre un. Bull. Soc. Roy. Sci. Liége 31, (1962) 751-756.
[Go3]Go3 Godeaux, L.: Sur les variétés algébriques à trois dimensions dont les sections hyperplanes sont des surfaces de bigenre un. Acad. Roy. Belg. Bull. Cl. Sci. 48, (1962) 1251-1257.
[Go4]Go4 Godeaux, L.: Une variété algébrique à trois dimensions à sections hyperplanes de bigenre un. Bull. Soc. Roy. Sci. Liége 39, (1970) 439-441.
[Gr]Gr Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differential Geom. 19, (1984) 125-171.
[H]H Hulek, K.: Projective geometry of elliptic curves. Astérisque 137 (1986).
[I1]I1 Iskovskih, V.A.: Fano 3-folds I. Izv. Akad. Nauk SSSR Ser. Mat. 41, (1977) 516-562, 717; English transl., Math. USSR-Izv. 11, (1977) 485–527.
[I2]I2 Iskovskih, V.A.: Fano 3-folds II. Izv. Akad. Nauk SSSR Ser. Mat. 42, (1978) 506-549; English transl., Math. USSR-Izv. 12, (1978) 469–506.
[I3]I3 Iskovskih, V.A.: Anticanonical models of three-dimensional algebraic varieties. Current problems in mathematics 12, VINITI, Moscow, 1979, 59-157, 239; English transl., J. Soviet Math. 13, (1980) 745–814.
[I4]I4 Iskovskih, V.A.: Birational automorphisms of three-dimensional algebraic varieties. Current problems in mathematics 12, VINITI, Moscow, 1979, 159-236, 239; English transl., J. Soviet Math. 13, (1980) 815–868.
[K]K Kleppe, J.O.: Liaison of families of subschemes in $\mathbb {P}^n$. In: Algebraic curves and projective geometry (Trento, 1988), Lecture Notes in Math. 1389, Springer, Berlin, 1989, 128-173.
[La]La Lazarsfeld, R.: A sampling of vector bundle techniques in the study of linear series. Lectures on Riemann surfaces (Trieste, 1987), World Sci. Publishing, Teaneck, NJ, 1989, 500-559.
[LS]LS Lange, H., Sernesi, E.: Quadrics containing a Prym-canonical curve. J. Algebraic Geom. 5, (1996) 387-399.
[Lv]Lv L’vovsky, S.: Extensions of projective varieties and deformations. I. Michigan Math. J. 39, (1992) 41-51.
[Ma]Ma Martens, G.: Über den Clifford-Index algebraischer Kurven. J. Reine Angew. Math. 336, (1982) 83-90.
[Mf]Mf Mumford, D.: Varieties defined by quadratic equations. In: Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, (1970) 29-100.
[MM]MM Mori, S., Mukai, S.: Classification of Fano $3$-folds with $B_ {2}\geq 2$. Manuscripta Math. 36, (1981/82) 147-162.
[Mu]Mu Murre, J.P.: Classification of Fano threefolds according to Fano and Iskovskih. In: Algebraic threefolds (Varenna, 1981), Lecture Notes in Math. 947, Springer, Berlin, 1982, 35-92.
[P]P Prokhorov, Y.G.: On three-dimensional varieties with hyperplane sections-Enriques surfaces. Mat. Sb. 186, (1995) 113-124; English transl., Sb. Math. 186, (1995) 1341–1352.
[R]R Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. of Math. 127, (1988) 309-316.
[Sa1]Sa1 Sano, T.: On classification of non-Gorenstein ${\mathbb Q}$-Fano $3$-folds of Fano index $1$. J. Math. Soc. Japan 47, (1995) 369-380.
[Sa2]Sa2 Sano, T.: Classification of non-Gorenstein ${\mathbb Q}$-Fano $d$-folds of Fano index greater than $d-2$. Nagoya Math. J. 142, (1996) 133-143.
[Sc]Sc Schreyer, F.O.: Syzygies of canonical curves and special linear series. Math. Ann. 275, (1986) 105-137.
[Sw]Sw Schwartau, P.W.: Liaison addition and monomial ideals. Ph.D. Thesis, Brandeis University, Waltham, MA, (1982).
[W1]W1 Wahl, J.: Introduction to Gaussian maps on an algebraic curve. In: Complex Projective Geometry, Trieste-Bergen 1989, London Math. Soc. Lecture Notes Ser. 179, Cambridge Univ. Press, Cambridge 1992, 304-323.
[W2]W2 Wahl, J.: Gaussian maps on algebraic curves. J. Differential Geom. 32, (1990) 77-98.
[Z]Z Zak, F.L.: Some properties of dual varieties and their application in projective geometry. In: Algebraic Geometry (Chicago, IL, 1989), Lecture Notes in Math. 1479, Springer, Berlin 1991, 273-280.
Additional Information
Luis Giraldo
Affiliation:
Departamento de Álgebra, Universidad Complutense de Madrid Avenida Complutense, s/n 28040 Madrid, Spain
Address at time of publication:
Departamento de Matematicas, Facultad de Ciencias, Universidad de Cadiz, Apartado 40, 11510 Puerto Real, Cadiz, Spain
Email:
luis.giraldo@uca.es
Angelo Felice Lopez
Affiliation:
Dipartimento di Matematica, Università di Roma Tre Largo San Leonardo Murialdo 1, 00146 Roma, Italy
MR Author ID:
289566
ORCID:
0000-0003-4923-6885
Email:
lopez@matrm3.mat.uniroma3.it
Roberto Muñoz
Affiliation:
ESCET, Universidad Rey Juan Carlos, 28933 Móstoles (Madrid), Spain
Email:
r.munoz@escet.urjc.es
Received by editor(s):
August 6, 2001
Published electronically:
September 17, 2003
Additional Notes:
The research of the first and third authors was partially supported by DGES research project, reference BFM2000-0621, and that of the second author was partially supported by the MURST national project “Geometria Algebrica"
