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010 a| 2023013008;

020 a| 9781470461577q| (paperback);

020 z| 9781470473532q| (pdf);

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084 a| 14B07a| 14C17a| 14C20a| 14E22a| 14H55a| 13C40a| 13H15a| 14H20a| 14M10a| 14M122| msc;

100 1 a| Patel, Anand P.?| UNAUTHORIZED;

245 10 a| Inflectionary invariants for isolated complete intersection curve singularitiesh| [electronic resource] /c| Anand P. Patel, Ashvin A. Swaminathan.;

260 a| Providence, RI :b| AMS, American Mathematical Society,c| 2023.;

300 a| vii, 99 pages ;c| 26 cm.;

490 0 a| Memoirs of the American Mathematical Society,x| 0065-9266 ;v| volume 282, number 1397;

504 a| Includes bibliographical references (pages 97-99).;

505 0 a| Background material -- Defining automatic degeneracy -- Automatic degeneracies of a node -- Automatic degeneracies of higher-order singularities -- Examples of computing automatic degeneracies -- Other enumerative applications.;

506 a| Available only to authorized users.;

520 a| "We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N [greater than or equal to] 2, and consider an isolated complete intersection curve singularity germ f : (CN, 0) [arrow] (CN[minus]1, 0). We define a numerical function m [x in a square] [arrow] ADm(2)(f) that naturally arises when counting mth-order weight-2 inflection points with ramification sequence (0, . . . , 0, 2) in a 1-parameter family of curves acquiring the singularity f [equals] 0, and we compute ADm(2)(f) for several interesting families of pairs (f,m). In particular, for a node defined by f : (x, y) [x in a square] [arrow] xy, we prove that ADm (2)(xy) [equals] [x in a square](m[plus]1)4 [x in a square], and we deduce as a corollary that ADm (2)(f) [greater than or equal to] (mult0 [delta symbol]f ) [x in a square](m[plus]1)4 [x in a square] for any f, where mult0 [delta symbol]f is the multiplicity of the discriminant [delta symbol]f at the origin in the deformation space. Significantly, we prove that the function m [x in a square] [arrow] ADm (2)(f)[minus](mult0 [delta symbol]f ) [x in a square](m[plus]1)4 [x in a square] is an analytic invariant measuring how much the singularity "counts as" an inflection point. We prove similar results for weight-2 inflection points with ramification sequence (0, . . . , 0, 1, 1) and for weight-1 inflection points, and we apply our results to solve a number of related enumerative problems"--c| Provided by publisher.;

538 a| Mode of access: World Wide Web;

650 0 a| Deformations of singularities.=| ^A539583;

650 0 a| Intersection theory (Mathematics)=| ^A190546;

650 0 a| Curves.=| ^A19286;

650 0 a| Invariants.=| ^A44671;

650 7 a| Algebraic geometry -- Local theory -- Deformations of singularities.2| msc;

650 7 a| Algebraic geometry -- Cycles and subschemes -- Intersection theory, characteristic classes, intersection multiplicities.2| msc;

650 7 a| Algebraic geometry -- Cycles and subschemes -- Divisors, linear systems, invertible sheaves.2| msc;

650 7 a| Algebraic geometry -- Birational geometry -- Ramification problems.2| msc;

650 7 a| Algebraic geometry -- Curves -- Riemann surfaces; Weierstrass points; gap sequences.2| msc;

650 7 a| Commutative algebra -- Theory of modules and ideals -- Linkage, complete intersections and determinantal ideals.2| msc;

650 7 a| Commutative algebra -- Local rings and semilocal rings -- Multiplicity theory and related topics.2| msc;

650 7 a| Algebraic geometry -- Curves -- Singularities, local rings.2| msc;

650 7 a| Algebraic geometry -- Special varieties -- Complete intersections.2| msc;

650 7 a| Algebraic geometry -- Special varieties -- Determinantal varieties.2| msc;

655 0 a| Electronic books.=| ^A491897;

700 1 a| Swaminathan, Ashvin A.?| UNAUTHORIZED;

856 40 z| Full text available from Ebook Central - Academic Completeu| https://ebookcentral.proquest.com/lib/eastcarolina/detail.action?docID=30404206;

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Inflectionary invariants for isolated complete intersection curve singularities / Anand P. Patel, Ashvin A. Swaminathan.

Author/creator	Patel, Anand P.
Other author	Swaminathan, Ashvin A.
Format	Electronic
Publication Info	Providence, RI : AMS, American Mathematical Society, 2023.
Description	vii, 99 pages ; 26 cm.
Supplemental Content	Full text available from Ebook Central - Academic Complete
Subjects	Deformations of singularities. Intersection theory (Mathematics) Curves. Invariants.

Series	Memoirs of the American Mathematical Society, 0065-9266 ; volume 282, number 1397
Contents	Background material -- Defining automatic degeneracy -- Automatic degeneracies of a node -- Automatic degeneracies of higher-order singularities -- Examples of computing automatic degeneracies -- Other enumerative applications.
Abstract	"We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N [greater than or equal to] 2, and consider an isolated complete intersection curve singularity germ f : (CN, 0) [arrow] (CN[minus]1, 0). We define a numerical function m [x in a square] [arrow] ADm(2)(f) that naturally arises when counting mth-order weight-2 inflection points with ramification sequence (0, . . . , 0, 2) in a 1-parameter family of curves acquiring the singularity f [equals] 0, and we compute ADm(2)(f) for several interesting families of pairs (f,m). In particular, for a node defined by f : (x, y) [x in a square] [arrow] xy, we prove that ADm (2)(xy) [equals] [x in a square](m[plus]1)4 [x in a square], and we deduce as a corollary that ADm (2)(f) [greater than or equal to] (mult0 [delta symbol]f ) [x in a square](m[plus]1)4 [x in a square] for any f, where mult0 [delta symbol]f is the multiplicity of the discriminant [delta symbol]f at the origin in the deformation space. Significantly, we prove that the function m [x in a square] [arrow] ADm (2)(f)[minus](mult0 [delta symbol]f ) [x in a square](m[plus]1)4 [x in a square] is an analytic invariant measuring how much the singularity "counts as" an inflection point. We prove similar results for weight-2 inflection points with ramification sequence (0, . . . , 0, 1, 1) and for weight-1 inflection points, and we apply our results to solve a number of related enumerative problems"-- Provided by publisher.
Bibliography note	Includes bibliographical references (pages 97-99).
Access restriction	Available only to authorized users.
Technical details	Mode of access: World Wide Web
Genre/form	Electronic books.
LCCN	2023013008
ISBN	9781470461577 (paperback)
ISBN	(pdf)

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Inflectionary invariants for isolated complete intersection curve singularities / Anand P. Patel, Ashvin A. Swaminathan.

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