Geometric theory of algebraic space curves:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1974
|
| Schriftenreihe: | Lecture notes in mathematics
423 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 302 S. |
| ISBN: | 3540069690 0387069690 |
Internformat
MARC
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| 245 | 1 | 0 | |a Geometric theory of algebraic space curves |c S. S. Abhyankar ; A. M. Sathaye |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1974 | |
| 300 | |a XIV, 302 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 423 | |
| 650 | 4 | |a Courbes algébriques supérieures | |
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| 650 | 7 | |a Géométrie différentielle |2 ram | |
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| 650 | 0 | 7 | |a Theorie |0 (DE-588)4059787-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Algebraische Geometrie |0 (DE-588)4001161-6 |2 gnd |9 rswk-swf |
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| 830 | 0 | |a Lecture notes in mathematics |v 423 |w (DE-604)BV000676446 |9 423 | |
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Datensatz im Suchindex
| _version_ | 1804117339487076352 |
|---|---|
| adam_text | CONTENTS
CHAPTER I. LOCAL GEOMETRY OR LENGTH., 1
§ 1. General terminology 1
§ 2 . Principal ideals and prime ideals . 1
§3. Total quotient ring and conductor 2
(3.1). Localization of the conductor.
§ 4. Normal model. 3
(4.1). Divisor of a function.
(4.2). Divisor of zeros of a function.
(4.3). The _ eifi = n formula.
§5. Length in a one-dimensional noetherian domain, or
affine intersection multiplicity on an irreducible
curve — 6
(5.1). Values of local intersection multiplicity.
Various cases.
(5.2). Local expansion of intersection multiplicity
over a divisor.
(5.3). Global intersection multiplicity.
(5.4). Local expansion of length.
(5.5). Intersection multiplicity equals length in
the integral closure (R ).
(5.6). Local intersection multiplicity equals a
length (in R) for a principal ideal.
(5.7). Globalization of (5.6).
(5.8). Special case of (5.6) - the normal case.
(5.9). Special case of (5.7) - the normal case.
(5.10). Definition. Multiplicity of a local domain of
dimension one.
(5.11). Definition and properties. Conductor, its
length; and adjoints.
(5.12). Lemma on overadjoints.
§6. Length in a one-dimensional noetherian homorphic _ 20
image, or affine intersection multiplicity on an
embedded irreducible curve.
(6.1). Values of local intersection multiplicity.
Various cases.
(6.2). Local expansion of intersection multiplicity
in the preimage.
(6.3). Global intersection multiplicity.
(6.4). Case of algebraically closed ground field
(6.5) . Case when a curve is thought to be
embedded in itself.
(6.6) to (6.9).
Restatements of (5.6) to (5.9) for the
case of a homomorphic image.
§ 7 . A commuting lemma for length 27
(7.1). For two embedded irreducible curves, at a
common simple point, the intersection multi¬
plicity of either one with the other is
the same.
(7.2). Globalization of (7.1) over a divisor.
VIII
(7.3) Complete globalization of (7.1).
§8. Length in a two-dimensional regular local domain . 28
(8.1). Intersection multiplicity of curves embedded
in a regular surface. Local case.
(8.2). For two curves embedded in a regular surface,
the intersection multiplicity of either one
with the other is the same. Local case.
(8.3). Additivity of intersection multiplicity of
curves embedded in a regular surface.
Local case.
§9. Multiplicity in a regular local domain. 30
(9.1). Multiplicity of an irreducible curve
(embedded in a regular surface) at a point
is the order of its defining equation.
(9.2). Technical lemma for (9.1).
§ 10. Double points of algebraic curves 33
(10.1). Theorem. Description of a double
point of a curve.
(10.2). Lemma. Description of high nodes.
(10.3). Lemma. Description of high cusps.
(10.4). Lemma. Description of nonrational
high cusps.
CHAPTER II. PROJECTIVE GEOMETRY OR HOMOGENEOUS DOMAINS 66
§11. Function fields and projective models. — 66
§ 12 . Homogeneous homomorphism. 68
§13. Homogeneous ideals (projective varieties)
and hypersurfaces. 68
§14. Homogeneous subdomains, flats (linear varieties),
projections, birational projections, and cones. 70
(14.1). Dimension and embedding dimension of
a homogeneous subdomain.
(14.2) and (14.3) .
Dimension and embedding dimension of a
(homogeneous) homomorphic image.
§15. Zeroset and homogeneous localization 75
(15.1), (15.2) and (15.3) .
Extension to (homogeneous) localization.
(15.4) and (15.5)
Alternative (affinized) description of
the (homogeneous) localization.
(15.6). Correspondence between homogeneous prime
ideals and homogeneous localization.
(15.7), (15.8) and (15.9).
Restatement of (15.1), (15.2) and (15.3)
for embedded varieties.
(15.10). Lemma. Number of conditions imposed on a
linear system of hypersurfaces.
§ 16 . Homogeneous coordinate systems . . 85
IX
§17. Polynomial rings as homogeneous domains. 86
(17.1) to (17.5).
Equivalent descriptions and properties of homo¬
geneous domains which are polynomial rings over
a field.
§18. Order on an embedded (irreducible) curve and
integral projections. 88
(18.1) and (18.2).
Order of a hypersurface at a valuation of an
embedded curve.
(18.3) and (18.4).
Order of an ideal at a valuation of an
embedded curve.
(18.5). Zerosets of ideals.
(18.6) to (18.10).
Order at a valuation of an embedded curve behaves
like a valualtion.
(18.11). Projection lemma. Projection of valuation
and order, from a vector space.
(18.12). Projection lemma. Projection of valuation
and order, from a flat (linear variety).
(18.13). Corollary-definition. Condition for a
tt-integral projection (where tt is a
hyperplane).
(18.13.1). Special case of (18.13)-projection from
a center not meeting the curve.
§19. Order on an abstract (irreducible) curve and
integral projections . 93
(19.1) to (19.12) .
Versions of (18.1) to (18.12) when a curve is
thought of as embedded in itself.
(19.13) and (19.13.1).
Versions of (18.13) and (18.13.1) for an
abstract curve.
(19.14). Remark. Integral ness of projection
commutes with homomorphic image.
§ 20 . Valued vector spaces . 95
(20.1) to (20.13).
Structure and properties of a valued
vector space.
§21. Osculating flats and integral projections of an
embedded (irreducible) curve. 109
(21.1) Definition and structure of osculating flats.
(21.2), (21.3) and (21.4) .
Application of §20 to the properties of osculating flats.
(21.5). Properties of osculating flats in special cases.
(21.6) . Condition for integral projection in terms of
osculating flats at the center of projections.
§22. Osculating flats and integral projections of an
abstract (irreducible) curve._ — 118
(22.1) to (22.6).
Restatements of (21.1) to (21.6) when a curve is
thought of as embedded in itself
(22.7). Remark. Osculating flats commute with
homomorphic image.
X
§23. Intersection multiplicity with an embedded . 121
(irreducible projective) curve.
(23.1), (23.2) and (23.3) .
Properties of intersection multiplicity with
an embedded curve.
(23.4). Case of algebraically closed ground field.
(23.5). Additivity of intersection multiplicity.
(23.6). Intersection multiplicity equals length
for a principal ideal.
(23.7). All points of an embedded line are simple.
(23.8) and (23.9). Bezout s Little Theorem.
Definition. Degree of an embedded
(irreducible) curve.
(23.10). Remark. Affine interpretation of degree.
(23.11). Lemma. If there are enough rational points
then embedding dimension of an embedded curve
is less than or equal to its degree.
(23.12) . Lemma. If the degree of an embedded curve
is one, then its embedding dimension is one.
(23.13). Remark. Hyperplanes have degree one.
(23.14). Projection formula. Projection of varieties
from flats.
(23.15). Special projection formula. Degree of
the projection.
(23.16). Remark. Case of an algebraically closed
ground field.
(23.17). Lemma. Birationality of the projection from
the generic point on a line.
(23.18). Definition and properties of tangents to
an embedded curve.
(23.19) and (23.20). Commuting lemmas. Versions of
(7.1) and (8.2) for projective curves.
§24. Intersection multiplicity with an abstract
(irreducible) curve. _ 138
(24.1) to (24.17). Versions of (23.1) to (23.17)
for an abstract curve.
(24.18). Definition and properties of tangents
to an abstract curve.
(24.19). Remark. Relations between an embedded
curve [A,C] and an abstract curve A/c.
(24.20). Lemma on overadjoints. Projective version
of (5.12) .
(24.21). Lemma on underadjoints. Existence of
certain type of projective underadjoints
which are true adjoints in an affine piece.
§25. Tangent cones and quasihyperplanes. 148
(25.1). Definition. Leading form of a hypersurface.
(25.2) and (25.3). Lemma-definition. Definition
and properties of tangent-cones.
(25.4). Definition. n-quasihyperplane.
(25.5). Lemma. Characterization of rr-quasihyperplanes.
(25.6). Lemma. A hyperplane (different from tt)
is a Tr-quasihyperplane.
(25.7). Lemma. Quadric n-quasiplanes.
(25.8). Lemma. Version of (9.1) for projective curves.
(25.9). Lemma. Degree of an embedded plane curve is
the degree of its defining equation.
XI
(25.10). Lemma. Characterization of tangent lines
of plane projective curves.
(25.11). Definition Intersection multiplicity of
two hypersurfaces.
(25.12). Additivity of the intersection multiplicity
of two hypersurfaces.
(25.13). Lemma. Version of (8.2) for projective
curves.
(25.14). Bezout s Theorem. Intersection of two
plane projective curves.
§26. 2-equimultiple plane projections of projective space
quintics . 155
(For notation see beginning of §26.)
(26.1). Lemma. Most lines through a d-fold point
of an irreducible curve are d-secants.
Also, if d degree of the curve, then
there are (d 11)-chords through the point
in every plane through the point.
(26.2). Lemma. An irreducible curve of degree 2 2
has 2-chords in every plane.
(26.3). Lemma. Sufficient condition for existance
of 4-chords.
(26.4). Lemma. Sufficient condition for existance
of 2-secants.
(26.5). Lemma. Projection from points on an
(n-l)-secant.
(26.6). Lemma. Projection from points on 2-secants
of an irreducible quartic.
(26.7). Lemma. Projection from points on certain
3-secants.
(26.8). Cone Lemma.
(26.9). Plane Lemma.
(26.10). Quadric Lemma.
(26.11). Proposition. Detailed description of pro¬
jections of curves of degree at most 5.
(26.12). Theorem. Condensed version of (26.11)
for reference.
CHAPTER III. BIRATIONAL GEOMETRY OR GENUS 180
§27. Different. 180
(27.1). Definitions.
(27.2). Dedekind s formula. § = £ dx .
(27.3). Lemma. Condition for Y
an unramified extension.
(27.4). Technical lemma.
(27.5). Lemma. Another characterization of an
unramified extension.
(27.6). Description of an unramified extension.
(27.7). Lemma. Finiteness of the set of
ramified primes.
(27.8). Definition. Separably generated
function fields.
(27.9). Lemma. Conditions for separably
generated function fields.
§28. Differentials. 189
(28.1). Reformulation of (?7-2) in terms of
divisors. Integral case.
XII
(28.2). Conversion formula for different.
Integral case.
(28.3) and (28.4). Exchange lemmas.
(28.5). General case of (28.2).
(28.6). General case of (28.1).
(28.7). Definition. ordv(ct,x) ; (a,x)
intended to be replaced by adx.
(28.8) to (28.10). Lemma. (a,x) behaves adx,
so far as or 3,, i-s concerned.
(28.11). Technical definition of genus.
(28.12). Theorem. A rational curve has genus zero
(28.13). Definition. Usual differentials and
their properties.
(28.14). Usual definition of genus.
(28.15). Genus formulas.
(28.16). Remark. Alternative proof to a
genus formula.
(28.17). Remark.
(28.18). Definition. Uniformizing parameter
and coordinate.
(28.19). Lemma. Properties of uniformizing
coordinates.
(28.20). Example. Valuations with unseparable
residue fields.
(28.21). Remark.
(28.22). Lemma. (28.19) reformulated using
differentials.
§ 29 . Genus of an abstract curve 220
(29.1) and (29.2). Genus formulas for plane
projective curves. Rationality of a
curve of genus zero.
(29.3). Remark.
(29.4). Direct proof of rationality of a conic.
(29.5). Direct proof of rationality of a line.
(29.6). Direct computation of the genus of a cubic.
(29.7). Theorem. Application of (24.21) to plane
projective curves of genus s 1 and degree 4
or 5.
§ 30 . Genus of an embedded curve. — 231
(30.1) and (30.2). Genus formulas. Rationality
of a curve of genus zero.
(30.3). Theorem. Combined version of (29.7)
and (26.12) for reference.
CHAPTER IV. AFFINE GEOMETRY OR FILTERED DOMAINS 234
§31. Filtered domains. Various definitions. 234
§32. Homogenization or taking projective completion. — 236
(32.1). Definition. Degree
(32.3) to (32.8). Properties of homogenization.
§33. Dehomogenization or taking an affine piece 238
(33.1). Definition. Dehomogenization
(33.2) to (33.4). Properties of dehomogenization
XIII
§34. Relation between homogenization and dehomogenization 241
§ 35 . Projection of a filtered domain. 244
(35.1). Definition. Projection.
(35.2). Relation with homogenization and
dehomogenization.
(35.3). Lemma. Conditions for integral projections.
(35.4). Definition. Degree, genus.
(35.5). Theorem. Affine version of (30.3) and
(26.12).
§ 36 . Complete intersections . 248
(36.1). Definition. Complete intersections,
essentially hyperplanar.
(36.2). Lemma. Essentially planar space curve
is a complete intersection.
(36.3). Corollary to (36.2).
(36.4). Lemma. A case of complete intersection.
(36.5). Theorem. A sufficient condition for
complete intersection.
(36.6). Elementary transformations.
(36.7). Theorem. Another sufficient condition
for complete intersection.
(36.8). Corollary to (36.7).
(36.9) . Main theorem of complete intersection.
CHAPTER V. APPENDIX. 257
§37. Double points of algebroid curves. An alternative
treatment of most of §10. 257
§38. Bezout s theorem. The general case — 268
§39. Chains of euclidean curves. A generalized
version of (36.7). 274
§40. Treatments of differentials in dimension one.
A short survey 280
§41. A generalization of Dedekind s formula about
conductor and different 281
§42 . The general adjoint condition. 285
§43. Geometric language. Geometric motivations
behind the various notations . 287
§ 44. Index to notations 295
§ 45 . Index to topics 298
Interdependence of sections.
In the following, §b «- §a1,...,§ar means §a1,...,§r are
directly referred to in §b. Except for such references, the only
other prerequisites for §b are the notations and definitions from
previous sections and they may be located from the index.
§i- §2, §3, §4, §5, basic.
|
| any_adam_object | 1 |
| author | Abhyankar, Shreeram Shankar 1930- Sathaye, Avinash M. |
| author_GND | (DE-588)119137569 |
| author_facet | Abhyankar, Shreeram Shankar 1930- Sathaye, Avinash M. |
| author_role | aut aut |
| author_sort | Abhyankar, Shreeram Shankar 1930- |
| author_variant | s s a ss ssa a m s am ams |
| building | Verbundindex |
| bvnumber | BV003050761 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SI 850 |
| ctrlnum | (OCoLC)612077715 (DE-599)BVBBV003050761 |
| dewey-full | 510/.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510/.8 |
| dewey-search | 510/.8 |
| dewey-sort | 3510 18 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV003050761 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:52:50Z |
| institution | BVB |
| isbn | 3540069690 0387069690 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001911418 |
| oclc_num | 612077715 |
| open_access_boolean | |
| owner | DE-12 DE-384 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-20 DE-824 DE-29T DE-19 DE-BY-UBM DE-210 DE-706 DE-83 DE-11 DE-188 |
| owner_facet | DE-12 DE-384 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-20 DE-824 DE-29T DE-19 DE-BY-UBM DE-210 DE-706 DE-83 DE-11 DE-188 |
| physical | XIV, 302 S. |
| psigel | HUB-ZB011200706 |
| publishDate | 1974 |
| publishDateSearch | 1974 |
| publishDateSort | 1974 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Abhyankar, Shreeram Shankar 1930- Verfasser (DE-588)119137569 aut Geometric theory of algebraic space curves S. S. Abhyankar ; A. M. Sathaye Berlin [u.a.] Springer 1974 XIV, 302 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 423 Courbes algébriques supérieures Courbes algébriques ram Géométrie différentielle ram Mathématiques Variétés algébriques ram Algebraische Kurve (DE-588)4001165-3 gnd rswk-swf Raumkurve (DE-588)4139969-9 gnd rswk-swf Theorie (DE-588)4059787-8 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 s Geometrie (DE-588)4020236-7 s Theorie (DE-588)4059787-8 s DE-604 Algebraische Geometrie (DE-588)4001161-6 s Raumkurve (DE-588)4139969-9 s Sathaye, Avinash M. Verfasser aut Lecture notes in mathematics 423 (DE-604)BV000676446 423 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001911418&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Abhyankar, Shreeram Shankar 1930- Sathaye, Avinash M. Geometric theory of algebraic space curves Lecture notes in mathematics Courbes algébriques supérieures Courbes algébriques ram Géométrie différentielle ram Mathématiques Variétés algébriques ram Algebraische Kurve (DE-588)4001165-3 gnd Raumkurve (DE-588)4139969-9 gnd Theorie (DE-588)4059787-8 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Geometrie (DE-588)4020236-7 gnd |
| subject_GND | (DE-588)4001165-3 (DE-588)4139969-9 (DE-588)4059787-8 (DE-588)4001161-6 (DE-588)4020236-7 |
| title | Geometric theory of algebraic space curves |
| title_auth | Geometric theory of algebraic space curves |
| title_exact_search | Geometric theory of algebraic space curves |
| title_full | Geometric theory of algebraic space curves S. S. Abhyankar ; A. M. Sathaye |
| title_fullStr | Geometric theory of algebraic space curves S. S. Abhyankar ; A. M. Sathaye |
| title_full_unstemmed | Geometric theory of algebraic space curves S. S. Abhyankar ; A. M. Sathaye |
| title_short | Geometric theory of algebraic space curves |
| title_sort | geometric theory of algebraic space curves |
| topic | Courbes algébriques supérieures Courbes algébriques ram Géométrie différentielle ram Mathématiques Variétés algébriques ram Algebraische Kurve (DE-588)4001165-3 gnd Raumkurve (DE-588)4139969-9 gnd Theorie (DE-588)4059787-8 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Geometrie (DE-588)4020236-7 gnd |
| topic_facet | Courbes algébriques supérieures Courbes algébriques Géométrie différentielle Mathématiques Variétés algébriques Algebraische Kurve Raumkurve Theorie Algebraische Geometrie Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001911418&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT abhyankarshreeramshankar geometrictheoryofalgebraicspacecurves AT sathayeavinashm geometrictheoryofalgebraicspacecurves |