Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Analysis on some infinite modules, inner projection, and applications

Authors: Kangjin Han and Sijong Kwak
Journal: Trans. Amer. Math. Soc. 364 (2012), 5791-5812
MSC (2010): Primary 14N05, 13D02, 14N25; Secondary 51N35
Published electronically: June 8, 2012
MathSciNet review: 2946932
Full-text PDF

Abstract | References | Similar Articles | Additional Information


A projective scheme $ X$ is called `quadratic' if $ X$ is scheme-theoretically cut out by homogeneous equations of degree $ 2$. Furthermore, we say that $ X$ satisfies `property $ \textbf {N}_{2,p}$' if it is quadratic and the quadratic ideal has only linear syzygies up to the first $ p$-th steps. In the present paper, we compare the linear syzygies of the inner projections with those of $ X$ and obtain a theorem on `embedded linear syzygies' as one of our main results. This is the natural projection-analogue of `restricting linear syzygies' in the linear section case. As an immediate corollary, we show that the inner projections of $ X$ satisfy property $ \textbf {N}_{2,p-1}$ for any reduced scheme $ X$ with property $ \textbf {N}_{2,p}$.

Moreover, we also obtain the neccessary lower bound $ (\operatorname {codim} X)\cdot p -\frac {p(p-1)}{2}$, which is sharp, on the number of quadrics vanishing on $ X$ in order to satisfy $ \textbf {N}_{2,p}$ and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme $ X$. These results admit an interesting `syzygetic' rigidity theorem on property $ \textbf {N}_{2,p}$ which leads the classifications of extremal and next-to-extremal 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.

References [Enhancements On Off] (What's this?)

  • [AK11] J. Ahn and S. Kwak, Graded mapping cone theorem, multisecants and syzygies, J. Algebra 331 (2011), 243-262. 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), 1-13. MR 1316568 (96a:14059)
  • [BHSS00] M. Beltrametti, A. Howard, M. Schneider and A. Sommese, Projections from subvarieties, Complex Analysis and Algebraic Geometry (T. Peternell, F-O. Schreyer, eds.), A volume in memory of Michael Schneider, (2000), 71-107. 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), 2033-2043. MR 2645336 (2012a:14120)
  • [BS07] M. Brodmann and P. Schenzel, Arithmetic properties of projective varieties of almost minimal degree, J. Algebraic Geom. 16 (2007), 347-400. 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, 97-106. MR 1823955 (2002b:14064)
  • [CKK06] Y. Choi, S. Kwak and P-L. Kang, Higher linear syzygies of inner projections, J. Algebra 305 (2006), 859-876. MR 2266857 (2007k:14008)
  • [CKP08] Y. Choi, S. Kwak and E. Park, On syzygies of non-complete embedding of projective varieties, Math. Zeitschrift 258, no. 2 (2008), 463-475. MR 2357647 (2009i:13023)
  • [EGHP05] D. Eisenbud, M. Green, K. Hulek and S. Popescu, Restriction linear syzygies: algebra and geometry, Compositio Math. 141 (2005), 1460-1478. 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, 1363-1389. 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, 573-605. 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, Springer-Verlag, Berlin, 1999. MR 1724388 (2001b:14010)
  • [Gre84] M. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), 125-171. MR 739785 (85e:14022)
  • [Gre98] M. Green, Generic Initial Ideals, in Six lectures on Commutative Algebra, (Elias J., Giral J.M., Miró-Roig, R.M., Zarzuela S., eds.), Progress in Mathematics 166, Birkhäuser, 1998, 119-186. 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, 301-314. 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., Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [Hoa93] L.T. Hoa, On minimal free resolutions of projective varieties of degree $ =$ codimension $ + 2$, J. Pure Appl. Algebra, 87 (1993), 241-250. MR 1228155 (94e:13024)
  • [Katz93] S. Katz, Arithmetically Cohen-Macaulay curves cut out by quadrics, Computational algebraic geometry and commutative algebra (Cortona, 1991), 257-263, Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993. MR 1253994 (94k:14021)
  • [KP05] S. Kwak and E. Park, Some effects of property $ {\rm N}\sb p$ on the higher normality and defining equations of nonlinearly normal varieties, J. Reine Angew. Math. 582 (2005), 87-105. MR 2139712 (2006d:14009)
  • [OP01] G. Ottaviani and R. Paoletti, Syzygies of Veronese embeddings, Compositio Math., 125 (2001), 31-37. 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.), 1-72.
  • [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), 425-426.
  • [Som79] A. 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 embedding, Math. Ann. 313 (1999), 507-545. MR 1678545 (2000b:14071)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14N05, 13D02, 14N25, 51N35

Retrieve articles in all journals with MSC (2010): 14N05, 13D02, 14N25, 51N35

Additional Information

Kangjin Han
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong, Yusung-Gu, Daejeon, Korea
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Korea

Sijong Kwak
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong, Yusung-Gu, Daejeon, Korea

Keywords: Linear syzygies, the mapping cone theorem, partial elimination ideals, inner projection, arithmetic depth, Castelnuovo-Mumford 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. 2009-0063180)
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.

American Mathematical Society