Varieties with one apparent double point
Authors:
Ciro Ciliberto, Massimiliano Mella and Francesco Russo
Journal:
J. Algebraic Geom. 13 (2004), 475-512
DOI:
https://doi.org/10.1090/S1056-3911-03-00355-2
Published electronically:
December 11, 2003
MathSciNet review:
2047678
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Additional Information
Abstract: The number of apparent double points of a smooth, irreducible projective variety $X$ of dimension $n$ in ${\mathbb P}^{2n+1}$ is the number of secant lines to $X$ passing through the general point of ${\mathbb P}^{2n+1}$. This classical notion dates back to Severi. In the present paper we classify smooth varieties of dimension at most three having one apparent double point. The techniques developed for this purpose allow us to treat a wider class of projective varieties.
[AB]ab M. Andreatta, E. Ballico, Classification of projective surfaces with small sectional genus: char $p\geq 0$, Rend. Sem. Mat. Univ. Padova 84 (1990), 175-193.
[AC]AC A. Arbarello, M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Ann. 256 (1981), 341-362.
[AR]AR A. Alzati, F. Russo, Special subhomaloidal systems of quadrics and varieties with one apparent double point, Math. Proc. Camb. Phil. Soc. 133 (2002).
[Br]Br J. Bronowski, The sum of powers as canonical expressions, Proc. Cam. Phil. Soc. 29 (1932), 69–82.
[CF]campflen F. Campana, H. Flenner, Projective threefolds containing a smooth rational surface with ample normal bundle, J. für die reine un angew. Math. 440 (1993), 77-98.
[CC1]cc1 L. Chiantini, C. Ciliberto, A few remarks on the lifting theorem, Asterisque 218 (1993), 95-109.
[CC2]cc L. Chiantini, C. Ciliberto, Threefolds with degenerate secant variety: on a theorem of G. Scorza, Marcel Dekker L.N. 217 (2001), 111–124.
[CC3]c2c L. Chiantini, C. Ciliberto, Weakly defective varieties, Trans. Amer. Math. Soc. 354 (2002), no. 1, 151–178.
[CS]cs C. Ciliberto, E. Sernesi, Singularities of the theta divisor and congruences of planes, J. Alg. Geom. 1 (1992), 231–250.
[DG]dg V. Di Gennaro, Alcune osservazioni sulle singolarità dello schema di Hilbert che parametrizza le varietà lineari contenute in una varietà proiettiva, Ricerche di Mat. 39 (1990), 259–291.
[Ed]Edge W. L. Edge, The number of apparent double points of certain loci, Proc. Cambridge Philos. Soc. 28 (1932), 285–299.
[Ei]Ei L. Ein, Varieties with small dual variety I, Invent. Math. 86 (1986), no. 1, 63–74.
[ES]ES L. Ein, N. Shepherd-Barron, Some special Cremona transformations, Amer. J. Math. 111 (1989), no. 5, 783–800.
[EH]EH D. Eisenbud, J. Harris, On varieties of minimal degree, Algebraic Geometry, Bowdoin 1985, Proc. Symp. in Pure Math. 46 (1987), 3–13.
[En]enr F. Enriques, Le superficie algebriche, Zanichelli, Bologna, (1949).
[F1]fr1 A. Franchetta, Sulla curva doppia della proiezione della superficie generale dell’$S_4$, da un punto generico su un $S_3$, Rend. Accad. d’Italia VII-2 (1941), 282-288.
[F2]fr2 A. Franchetta, Sulla curva doppia della proiezione della superficie generale dell’$S_4$, da un punto generico su un $S_3$, Rend. Accad. Naz. Lincei VIII-2 (1947), 276-279.
[GH]GH Ph. Griffiths, J. Harris, Algebraic geometry and local differential geometry, Ann. Scient. Ec. Norm. Sup. 4 12 (1979), 355-432.
[Ha]ha R. Hartshorne, Algebraic Geometry, Springer Graduate Texts in Math., (1977).
[Hi]hi H. Hironaka, On the arithmetic genera and the effective genera of algebraic curves, Mem. Coll. Sci. Univ. Kyoto A 30 (1957), 177-195.
[HKS]HKS K. Hulek, S. Katz, F.O. Schreyer, Cremona transformations and syzygies, Math. Z. 209 (1992), 419–443.
[Io1]Io1 P. Ionescu, Embedded projective varieties of small invariants, Springer L.N.M. 1056 (1984), 142–186.
[Io2]Io3 P. Ionescu, On varieties whose degree is small with respect to codimension, Math. Ann. 271 (1985), 339–348.
[Io3]Io5 P. Ionescu, Generalized adjunction and applications, Math. Proc. Camb. Phil. Soc. 99 (1986), 457–472.
[Io4]Io4 P. Ionescu, Embedded projective varieties of small invariants. II, Rev. Roum. Math. 31 (1986), 539–545.
[Io5]Io2 P. Ionescu, Embedded projective varieties of small invariants, III, Springer L.N.M. 1417 (1990), 138–154.
[Is1]Is1 V. A. Iskovskikh, Fano 3-folds I, Math. USSR Izvestija, AMS Translations 11 (1977), 485–527.
[Is2]Is2 V. A. Iskovskikh, Fano 3-folds II, Math. USSR Izvestija, AMS Translations 12 (1978), 469–506.
[Kl]Kl S. Kleiman, About the conormal scheme, in Complete intersections (Acireale, 1983), Lecture Notes in Math. 1092, Springer, Berlin, (1984), 161–197.
[LP]LP J. Le Potier, Stabilité et amplitude sur ${\mathbb P}_2(\textbf {C})$. (French) Vector bundles and differential equations (Proc. Conf., Nice, 1979), pp. 145–182, Progr. Math. 7, Birkhäuser (1980), 145–182.
[Li]li L. Livorni, Classification of algebraic varieties with sectional genus less than or equal to six. I: rational surfaces, Pacific J. of Math. 113 (1984), 93-114.
[LV]LV R. Lazarsfeld, A. van de Ven, Topics in the geometry of projective space, DMV Seminar 4, Birkhäuser, (1984).
[Ma]mattuck A. Mattuck, On the symmetric product of a rational surface, Proc. Amer. Math. Soc. 21 (1969), 683-688.
[Me]Me M. Mella, Existence of good divisors on Mukai varieties, J. of Alg. Geom. 8 (1999), 197–206.
[MR]MR M. Mella, F. Russo, On Cremona transformations whose base locus is a smooth irreducible surface, pre-print (2001).
[Mez]mez E. Mezzetti, Projective varieties with many degenerate subvarieties, Boll. Un. Mat. Ital. B (7) 8 (1994), no. 4, 807–832.
[MP]MP E. Mezzetti, D. Portelli, Threefolds in ${\mathbb P}^5$ with a $3$-dimensional family of plane curves, Manuscripta Math. 90 (1996), 365–381.
[Mo]mori S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math. 116 (1982), 133–176.
[Mu1]Mu S. Mukai, Biregular classification of Fano threefolds and Fano manifolds of coindex 3, Proc. Nat. Acad. Sci. USA 86 (1989), 3000–3002.
[Mu2]Mu2 S. Mukai, Simple Lie algebra and Legendre variety, preprint (1998) http://www.math.nagoya-u.ac.jp/˜mukai
[Pi]Pi R. Piene, A proof of Noether’s formula for the arithmetic genus of an algebraic surface, Compositio Math. 38 (1979), 113-119.
[Ro]rogora E. Rogora, Medodi proiettivi per lo studio di alcune questioni relative alle varietà immerse, Tesi di Dottorato, Università di Roma “La Sapienza", (1993).
[Ru]Ru F. Russo, On a theorem of Severi, Math. Ann. 316 (2000), 1–17.
[ST]ST2 J. G. Semple, J. A. Tyrrell, The $T_{2,4}$ of $S_6$ defined by a rational surface $F^8$, Proc. Lond. Math. Soc. 20 (1970), 205–221.
[Se]Se F. Severi, Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni e ai suoi punti tripli apparenti, Rend. Circ. Mat. Palermo 15 (1901), 33–51.
[T]terracini A. Terracini Sulle $V_k$ che rappresentano più di ${{k(k-1)}/2}$ equazioni di Laplace linearmente indipendenti, Rend. Circ. Mat. Palermo, 33 (1912).
[vdV]vdv A. van de Ven, On the $2$-connectedness of very ample divisors on a surface, Duke Math. J. 46 (1979), no. 2, 403–407.
[V]violo M. G. Violo, Varietà con un punto doppio apparente, Tesi di Dottorato, Torino, (1997).
[Za1]Zak1 F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs 127, Amer. Math. Soc. (1993).
[Za2]Zak2 F. L. Zak, Varieties of small codimension arising from group actions, Addendum to: [LV].
[AB]ab M. Andreatta, E. Ballico, Classification of projective surfaces with small sectional genus: char $p\geq 0$, Rend. Sem. Mat. Univ. Padova 84 (1990), 175-193.
[AC]AC A. Arbarello, M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Ann. 256 (1981), 341-362.
[AR]AR A. Alzati, F. Russo, Special subhomaloidal systems of quadrics and varieties with one apparent double point, Math. Proc. Camb. Phil. Soc. 133 (2002).
[Br]Br J. Bronowski, The sum of powers as canonical expressions, Proc. Cam. Phil. Soc. 29 (1932), 69–82.
[CF]campflen F. Campana, H. Flenner, Projective threefolds containing a smooth rational surface with ample normal bundle, J. für die reine un angew. Math. 440 (1993), 77-98.
[CC1]cc1 L. Chiantini, C. Ciliberto, A few remarks on the lifting theorem, Asterisque 218 (1993), 95-109.
[CC2]cc L. Chiantini, C. Ciliberto, Threefolds with degenerate secant variety: on a theorem of G. Scorza, Marcel Dekker L.N. 217 (2001), 111–124.
[CC3]c2c L. Chiantini, C. Ciliberto, Weakly defective varieties, Trans. Amer. Math. Soc. 354 (2002), no. 1, 151–178.
[CS]cs C. Ciliberto, E. Sernesi, Singularities of the theta divisor and congruences of planes, J. Alg. Geom. 1 (1992), 231–250.
[DG]dg V. Di Gennaro, Alcune osservazioni sulle singolarità dello schema di Hilbert che parametrizza le varietà lineari contenute in una varietà proiettiva, Ricerche di Mat. 39 (1990), 259–291.
[Ed]Edge W. L. Edge, The number of apparent double points of certain loci, Proc. Cambridge Philos. Soc. 28 (1932), 285–299.
[Ei]Ei L. Ein, Varieties with small dual variety I, Invent. Math. 86 (1986), no. 1, 63–74.
[ES]ES L. Ein, N. Shepherd-Barron, Some special Cremona transformations, Amer. J. Math. 111 (1989), no. 5, 783–800.
[EH]EH D. Eisenbud, J. Harris, On varieties of minimal degree, Algebraic Geometry, Bowdoin 1985, Proc. Symp. in Pure Math. 46 (1987), 3–13.
[En]enr F. Enriques, Le superficie algebriche, Zanichelli, Bologna, (1949).
[F1]fr1 A. Franchetta, Sulla curva doppia della proiezione della superficie generale dell’$S_4$, da un punto generico su un $S_3$, Rend. Accad. d’Italia VII-2 (1941), 282-288.
[F2]fr2 A. Franchetta, Sulla curva doppia della proiezione della superficie generale dell’$S_4$, da un punto generico su un $S_3$, Rend. Accad. Naz. Lincei VIII-2 (1947), 276-279.
[GH]GH Ph. Griffiths, J. Harris, Algebraic geometry and local differential geometry, Ann. Scient. Ec. Norm. Sup. 4 12 (1979), 355-432.
[Ha]ha R. Hartshorne, Algebraic Geometry, Springer Graduate Texts in Math., (1977).
[Hi]hi H. Hironaka, On the arithmetic genera and the effective genera of algebraic curves, Mem. Coll. Sci. Univ. Kyoto A 30 (1957), 177-195.
[HKS]HKS K. Hulek, S. Katz, F.O. Schreyer, Cremona transformations and syzygies, Math. Z. 209 (1992), 419–443.
[Io1]Io1 P. Ionescu, Embedded projective varieties of small invariants, Springer L.N.M. 1056 (1984), 142–186.
[Io2]Io3 P. Ionescu, On varieties whose degree is small with respect to codimension, Math. Ann. 271 (1985), 339–348.
[Io3]Io5 P. Ionescu, Generalized adjunction and applications, Math. Proc. Camb. Phil. Soc. 99 (1986), 457–472.
[Io4]Io4 P. Ionescu, Embedded projective varieties of small invariants. II, Rev. Roum. Math. 31 (1986), 539–545.
[Io5]Io2 P. Ionescu, Embedded projective varieties of small invariants, III, Springer L.N.M. 1417 (1990), 138–154.
[Is1]Is1 V. A. Iskovskikh, Fano 3-folds I, Math. USSR Izvestija, AMS Translations 11 (1977), 485–527.
[Is2]Is2 V. A. Iskovskikh, Fano 3-folds II, Math. USSR Izvestija, AMS Translations 12 (1978), 469–506.
[Kl]Kl S. Kleiman, About the conormal scheme, in Complete intersections (Acireale, 1983), Lecture Notes in Math. 1092, Springer, Berlin, (1984), 161–197.
[LP]LP J. Le Potier, Stabilité et amplitude sur ${\mathbb P}_2(\textbf {C})$. (French) Vector bundles and differential equations (Proc. Conf., Nice, 1979), pp. 145–182, Progr. Math. 7, Birkhäuser (1980), 145–182.
[Li]li L. Livorni, Classification of algebraic varieties with sectional genus less than or equal to six. I: rational surfaces, Pacific J. of Math. 113 (1984), 93-114.
[LV]LV R. Lazarsfeld, A. van de Ven, Topics in the geometry of projective space, DMV Seminar 4, Birkhäuser, (1984).
[Ma]mattuck A. Mattuck, On the symmetric product of a rational surface, Proc. Amer. Math. Soc. 21 (1969), 683-688.
[Me]Me M. Mella, Existence of good divisors on Mukai varieties, J. of Alg. Geom. 8 (1999), 197–206.
[MR]MR M. Mella, F. Russo, On Cremona transformations whose base locus is a smooth irreducible surface, pre-print (2001).
[Mez]mez E. Mezzetti, Projective varieties with many degenerate subvarieties, Boll. Un. Mat. Ital. B (7) 8 (1994), no. 4, 807–832.
[MP]MP E. Mezzetti, D. Portelli, Threefolds in ${\mathbb P}^5$ with a $3$-dimensional family of plane curves, Manuscripta Math. 90 (1996), 365–381.
[Mo]mori S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math. 116 (1982), 133–176.
[Mu1]Mu S. Mukai, Biregular classification of Fano threefolds and Fano manifolds of coindex 3, Proc. Nat. Acad. Sci. USA 86 (1989), 3000–3002.
[Mu2]Mu2 S. Mukai, Simple Lie algebra and Legendre variety, preprint (1998) http://www.math.nagoya-u.ac.jp/˜mukai
[Pi]Pi R. Piene, A proof of Noether’s formula for the arithmetic genus of an algebraic surface, Compositio Math. 38 (1979), 113-119.
[Ro]rogora E. Rogora, Medodi proiettivi per lo studio di alcune questioni relative alle varietà immerse, Tesi di Dottorato, Università di Roma “La Sapienza", (1993).
[Ru]Ru F. Russo, On a theorem of Severi, Math. Ann. 316 (2000), 1–17.
[ST]ST2 J. G. Semple, J. A. Tyrrell, The $T_{2,4}$ of $S_6$ defined by a rational surface $F^8$, Proc. Lond. Math. Soc. 20 (1970), 205–221.
[Se]Se F. Severi, Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni e ai suoi punti tripli apparenti, Rend. Circ. Mat. Palermo 15 (1901), 33–51.
[T]terracini A. Terracini Sulle $V_k$ che rappresentano più di ${{k(k-1)}/2}$ equazioni di Laplace linearmente indipendenti, Rend. Circ. Mat. Palermo, 33 (1912).
[vdV]vdv A. van de Ven, On the $2$-connectedness of very ample divisors on a surface, Duke Math. J. 46 (1979), no. 2, 403–407.
[V]violo M. G. Violo, Varietà con un punto doppio apparente, Tesi di Dottorato, Torino, (1997).
[Za1]Zak1 F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs 127, Amer. Math. Soc. (1993).
[Za2]Zak2 F. L. Zak, Varieties of small codimension arising from group actions, Addendum to: [LV].
Additional Information
Ciro Ciliberto
Affiliation:
Dipartimento di Matematica, Universitá di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italia
MR Author ID:
49480
Email:
cilibert@axp.mat.uniroma2.it
Massimiliano Mella
Affiliation:
Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italia
Email:
mll@unife.it
Francesco Russo
Affiliation:
Departamento de Matematica, Universidade Federal de Pernambuco, Cidade Universitaria, 50670-901 Recife–PE, Brasil
MR Author ID:
214972
ORCID:
0000-0002-5889-783X
Email:
frusso@dmat.ufpe.br
Received by editor(s):
January 24, 2002
Published electronically:
December 11, 2003
