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

DOI:
https://doi.org/10.1090/S0002-9947-2012-05755-2

Published electronically:
June 8, 2012

MathSciNet review:
2946932

Full-text PDF

Abstract | References | Similar Articles | Additional Information

A projective scheme is called `quadratic' if is scheme-theoretically 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 projection-analogue 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 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.

**[AK11]**Jeaman Ahn and Sijong Kwak,*Graded mapping cone theorem, multisecants and syzygies*, J. Algebra**331**(2011), 243–262. MR**2774656**, https://doi.org/10.1016/j.jalgebra.2010.07.030**[AS10]**A. Alzati and J.C. Sierra,*A bound on the degree of schemes defined by quadratic equations*, to appear in `Forum Mathematicum'.**[Bau95]**Ingrid Bauer,*Inner projections of algebraic surfaces: a finiteness result*, J. Reine Angew. Math.**460**(1995), 1–13. MR**1316568**, https://doi.org/10.1515/crll.1995.460.1**[BHSS00]**Mauro C. Beltrametti, Alan Howard, Michael Schneider, and Andrew J. Sommese,*Projections from subvarieties*, Complex analysis and algebraic geometry, de Gruyter, Berlin, 2000, pp. 71–107. MR**1760873****[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), no. 11, 2033–2043. MR**2645336**, https://doi.org/10.1016/j.jpaa.2010.02.009**[BS07]**Markus Brodmann and Peter Schenzel,*Arithmetic properties of projective varieties of almost minimal degree*, J. Algebraic Geom.**16**(2007), no. 2, 347–400. MR**2274517**, https://doi.org/10.1090/S1056-3911-06-00442-5**[CC01]**Alberto 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**, https://doi.org/10.1515/advg.2001.007**[CKK06]**Youngook Choi, Pyung-Lyun Kang, and Sijong Kwak,*Higher linear syzygies of inner projections*, J. Algebra**305**(2006), no. 2, 859–876. MR**2266857**, https://doi.org/10.1016/j.jalgebra.2006.08.007**[CKP08]**Youngook Choi, Sijong Kwak, and Euisung Park,*On syzygies of non-complete embedding of projective varieties*, Math. Z.**258**(2008), no. 2, 463–475. MR**2357647**, https://doi.org/10.1007/s00209-007-0181-9**[EGHP05]**David Eisenbud, Mark Green, Klaus Hulek, and Sorin Popescu,*Restricting linear syzygies: algebra and geometry*, Compos. Math.**141**(2005), no. 6, 1460–1478. MR**2188445**, https://doi.org/10.1112/S0010437X05001776**[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****[EHU06]**David Eisenbud, Craig Huneke, and Bernd Ulrich,*The regularity of Tor and graded Betti numbers*, Amer. J. Math.**128**(2006), no. 3, 573–605. MR**2230917****[Fuj90]**Takao Fujita,*Classification theories of polarized varieties*, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. MR**1162108****[FCV99]**H. Flenner, L. O’Carroll, and W. Vogel,*Joins and intersections*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR**1724388****[Gre84]**Mark L. Green,*Koszul cohomology and the geometry of projective varieties*, J. Differential Geom.**19**(1984), no. 1, 125–171. MR**739785****[Gre98]**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****[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****[HK10]**K. Han and S. Kwak,*Projections from lines: algebraic and geometric properties*, in preparation.**[Hart77]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[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**, https://doi.org/10.1016/0022-4049(93)90112-7**[Katz93]**Sheldon Katz,*Arithmetically Cohen-Macaulay 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****[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**, https://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**, https://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.), 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]**Andrew John Sommese,*Hyperplane sections of projective surfaces. I. The adjunction mapping*, Duke Math. J.**46**(1979), no. 2, 377–401. MR**534057****[Zak99]**F. L. Zak,*Projective invariants of quadratic embeddings*, Math. Ann.**313**(1999), no. 3, 507–545. MR**1678545**, https://doi.org/10.1007/s002080050271

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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

Email:
han.kangjin@kaist.ac.kr, kangjin.han@kias.re.kr

**Sijong Kwak**

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

Email:
skwak@kaist.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-2012-05755-2

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.