Analysis on some infinite modules, inner projection, and applications
Authors:
Kangjin Han and Sijong Kwak
Journal:
Trans. Amer. Math. Soc. 364 (2012), 57915812
MSC (2010):
Primary 14N05, 13D02, 14N25; Secondary 51N35
Published electronically:
June 8, 2012
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Abstract: A projective scheme is called `quadratic' if is schemetheoretically cut out by homogeneous equations of degree . Furthermore, we say that satisfies `property ' if it is quadratic and the quadratic ideal has only linear syzygies up to the first th steps. In the present paper, we compare the linear syzygies of the inner projections with those of and obtain a theorem on `embedded linear syzygies' as one of our main results. This is the natural projectionanalogue of `restricting linear syzygies' in the linear section case. As an immediate corollary, we show that the inner projections of satisfy property for any reduced scheme with property . Moreover, we also obtain the neccessary lower bound , which is sharp, on the number of quadrics vanishing on in order to satisfy and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme . These results admit an interesting `syzygetic' rigidity theorem on property which leads the classifications of extremal and nexttoextremal cases. For these results we develop the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by M. Green. This new method allows us to treat a wider class of projective schemes which cannot be covered by the Koszul cohomology techniques because these are not projectively normal in general.
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Jeaman
Ahn and Sijong
Kwak, Graded mapping cone theorem, multisecants and syzygies,
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(2012f:14103), http://dx.doi.org/10.1016/j.jalgebra.2010.07.030
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MR
1316568 (96a:14059), http://dx.doi.org/10.1515/crll.1995.460.1
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1760873 (2001e:14004)
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Calabri and Ciro
Ciliberto, On special projections of varieties: epitome to a
theorem of Beniamino Segre, Adv. Geom. 1 (2001),
no. 1, 97–106. MR 1823955
(2002b:14064), http://dx.doi.org/10.1515/advg.2001.007
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Youngook
Choi, PyungLyun
Kang, and Sijong
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Youngook
Choi, Sijong
Kwak, and Euisung
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Eisenbud, M.
Green, K.
Hulek, and S.
Popescu, Small schemes and varieties of minimal degree, Amer.
J. Math. 128 (2006), no. 6, 1363–1389. MR 2275024
(2007j:14078)
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David
Eisenbud, Craig
Huneke, and Bernd
Ulrich, The regularity of Tor and graded Betti numbers, Amer.
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(2007b:13027)
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H.
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Mathematics, SpringerVerlag, Berlin, 1999. MR 1724388
(2001b:14010)
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Mark
L. Green, Koszul cohomology and the geometry of projective
varieties, J. Differential Geom. 19 (1984),
no. 1, 125–171. MR 739785
(85e:14022)
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Mark
L. Green, Generic initial ideals, Six lectures on commutative
algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser,
Basel, 1998, pp. 119–186. MR 1648665
(99m:13040)
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M.
Green and R.
Lazarsfeld, Some results on the syzygies of finite sets and
algebraic curves, Compositio Math. 67 (1988),
no. 3, 301–314. MR 959214
(90d:14034)
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K. Han and S. Kwak, Projections from lines: algebraic and geometric properties, in preparation.
 [Hart77]
Robin
Hartshorne, Algebraic geometry, SpringerVerlag, New
YorkHeidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
(57 #3116)
 [Hoa93]
Le
Tuan Hoa, On minimal free resolutions of projective varieties of
degree
=𝑐𝑜𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛+2,
J. Pure Appl. Algebra 87 (1993), no. 3,
241–250. MR 1228155
(94e:13024), http://dx.doi.org/10.1016/00224049(93)901127
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Sheldon
Katz, Arithmetically CohenMacaulay curves cut out by
quadrics, Computational algebraic geometry and commutative algebra
(Cortona, 1991), Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge,
1993, pp. 257–263. MR 1253994
(94k:14021)
 [KP05]
Sijong
Kwak and Euisung
Park, Some effects of property 𝑁_{𝑝} on the higher
normality and defining equations of nonlinearly normal varieties, J.
Reine Angew. Math. 582 (2005), 87–105. MR 2139712
(2006d:14009), http://dx.doi.org/10.1515/crll.2005.2005.582.87
 [OP01]
Giorgio
Ottaviani and Raffaella
Paoletti, Syzygies of Veronese embeddings, Compositio Math.
125 (2001), no. 1, 31–37. MR 1818055
(2002g:13023), http://dx.doi.org/10.1023/A:1002662809474
 [Park08]
E. Park, On secant loci and simple linear projections of some projective varieties, preprint
 [Reid00]
M. Reid, Graded rings and birational geometry, in Proc. of Algebraic Geometry Symposium (Kinosaki, Oct. 2000), K. Ohno (Ed.), 172.
 [Seg36]
B. Segre, On the locus of points from which an algebraic variety is projected multiply, Proc. Phys.Math. Soc. Japan Ser. III, 18 (1936), 425426.
 [Som79]
Andrew
John Sommese, Hyperplane sections of projective surfaces. I. The
adjunction mapping, Duke Math. J. 46 (1979),
no. 2, 377–401. MR 534057
(82f:14033)
 [Zak99]
F.
L. Zak, Projective invariants of quadratic embeddings, Math.
Ann. 313 (1999), no. 3, 507–545. MR 1678545
(2000b:14071), http://dx.doi.org/10.1007/s002080050271
 [AK11]
 J. Ahn and S. Kwak, Graded mapping cone theorem, multisecants and syzygies, J. Algebra 331 (2011), 243262. MR 2774656
 [AS10]
 A. Alzati and J.C. Sierra, A bound on the degree of schemes defined by quadratic equations, to appear in `Forum Mathematicum'.
 [Bau95]
 I. Bauer, Inner projections of algebraic surfaces: a finiteness result, J. Reine Angew. Math. 460 (1995), 113. MR 1316568 (96a:14059)
 [BHSS00]
 M. Beltrametti, A. Howard, M. Schneider and A. Sommese, Projections from subvarieties, Complex Analysis and Algebraic Geometry (T. Peternell, FO. Schreyer, eds.), A volume in memory of Michael Schneider, (2000), 71107. MR 1760873 (2001e:14004)
 [BP10]
 M. Brodmann and E. Park, On varieties of almost minimal degree I: Secant loci of rational normal scrolls, J. Pure Appl. Algebra 214 (2010), 20332043. MR 2645336 (2012a:14120)
 [BS07]
 M. Brodmann and P. Schenzel, Arithmetic properties of projective varieties of almost minimal degree, J. Algebraic Geom. 16 (2007), 347400. MR 2274517 (2008b:14085)
 [CC01]
 A. Calabri and C. Ciliberto, On special projections of varieties: epitome to a theorem of Beniamino Segre, Adv. Geom. 1 (2001), no. 1, 97106. MR 1823955 (2002b:14064)
 [CKK06]
 Y. Choi, S. Kwak and PL. Kang, Higher linear syzygies of inner projections, J. Algebra 305 (2006), 859876. MR 2266857 (2007k:14008)
 [CKP08]
 Y. Choi, S. Kwak and E. Park, On syzygies of noncomplete embedding of projective varieties, Math. Zeitschrift 258, no. 2 (2008), 463475. MR 2357647 (2009i:13023)
 [EGHP05]
 D. Eisenbud, M. Green, K. Hulek and S. Popescu, Restriction linear syzygies: algebra and geometry, Compositio Math. 141 (2005), 14601478. MR 2188445 (2006m:14072)
 [EGHP06]
 D. Eisenbud, M. Green, K. Hulek and S. Popescu, Small schemes and varieties of minimal degree, Amer. J. Math. 128 (2006), no. 6, 13631389. MR 2275024 (2007j:14078)
 [EHU06]
 D. Eisenbud, C. Huneke, and B. Ulrich, The regularity of Tor and graded Betti numbers, Amer. J. Math. 128 (2006), no. 3, 573605. MR 2230917 (2007b:13027)
 [Fuj90]
 T. Fujita, Classification theories of polarized varieties, Cambridge University Press, Cambridge, 1990. MR 1162108 (93e:14009)
 [FCV99]
 H. Flenner, L. O'Carroll, and W. Vogel, Joins and intersections, SpringerVerlag, Berlin, 1999. MR 1724388 (2001b:14010)
 [Gre84]
 M. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), 125171. MR 739785 (85e:14022)
 [Gre98]
 M. Green, Generic Initial Ideals, in Six lectures on Commutative Algebra, (Elias J., Giral J.M., MiróRoig, R.M., Zarzuela S., eds.), Progress in Mathematics 166, Birkhäuser, 1998, 119186. MR 1648665 (99m:13040)
 [GL88]
 M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988), no. 3, 301314. MR 959214 (90d:14034)
 [HK10]
 K. Han and S. Kwak, Projections from lines: algebraic and geometric properties, in preparation.
 [Hart77]
 R. Hartshorne, Algebraic geometry, Graduate Texts in Math., SpringerVerlag, New York, 1977. MR 0463157 (57:3116)
 [Hoa93]
 L.T. Hoa, On minimal free resolutions of projective varieties of degree codimension , J. Pure Appl. Algebra, 87 (1993), 241250. MR 1228155 (94e:13024)
 [Katz93]
 S. Katz, Arithmetically CohenMacaulay curves cut out by quadrics, Computational algebraic geometry and commutative algebra (Cortona, 1991), 257263, Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993. MR 1253994 (94k:14021)
 [KP05]
 S. Kwak and E. Park, Some effects of property on the higher normality and defining equations of nonlinearly normal varieties, J. Reine Angew. Math. 582 (2005), 87105. MR 2139712 (2006d:14009)
 [OP01]
 G. Ottaviani and R. Paoletti, Syzygies of Veronese embeddings, Compositio Math., 125 (2001), 3137. MR 1818055 (2002g:13023)
 [Park08]
 E. Park, On secant loci and simple linear projections of some projective varieties, preprint
 [Reid00]
 M. Reid, Graded rings and birational geometry, in Proc. of Algebraic Geometry Symposium (Kinosaki, Oct. 2000), K. Ohno (Ed.), 172.
 [Seg36]
 B. Segre, On the locus of points from which an algebraic variety is projected multiply, Proc. Phys.Math. Soc. Japan Ser. III, 18 (1936), 425426.
 [Som79]
 A. Sommese, Hyperplane sections of projective surfaces I, The adjunction mapping, Duke Math. J. 46 (1979), no. 2, 377401. MR 534057 (82f:14033)
 [Zak99]
 F. L. Zak, Projective invariants of quadratic embedding, Math. Ann. 313 (1999), 507545. MR 1678545 (2000b:14071)
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Additional Information
Kangjin Han
Affiliation:
Department of Mathematics, Korea Advanced Institute of Science and Technology, 3731 Gusungdong, YusungGu, Daejeon, Korea
Address at time of publication:
School of Mathematics, Korea Institute for Advanced Study, Seoul 130722, Korea
Email:
han.kangjin@kaist.ac.kr, kangjin.han@kias.re.kr
Sijong Kwak
Affiliation:
Department of Mathematics, Korea Advanced Institute of Science and Technology, 3731 Gusungdong, YusungGu, Daejeon, Korea
Email:
skwak@kaist.ac.kr
DOI:
http://dx.doi.org/10.1090/S000299472012057552
PII:
S 00029947(2012)057552
Keywords:
Linear syzygies,
the mapping cone theorem,
partial elimination ideals,
inner projection,
arithmetic depth,
CastelnuovoMumford regularity
Received by editor(s):
October 14, 2010
Published electronically:
June 8, 2012
Additional Notes:
The authors were supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant No. 20090063180)
Dedicated:
Dedicated to the memory of Hyo Chul Myung (June, 1937–February, 2010)
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
