Geometry of Algebraic Curves

Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip; Harris, Joe

doi:10.1007/978-1-4757-5323-3

Cited by 1,498 publications

(1,382 citation statements)

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“…k (u g ) the kth complete symmetric function, namely the sum of all monomials of total degree k of the variables u (1) g , . .…”

Section: The Schur-weierstrass Polynomialmentioning

confidence: 99%

“…We call this the Sato weight. It is easy to see that S(u) is homogeneous with respect to the Sato weight, and has the Sato weight 1 2 g(g + 1). Let m be a fixed positive integer and ξ 1 , .…”

Section: This Polynomial S(u) Is Called the Schur-weierstrass Polynommentioning

confidence: 99%

“…· · · (n − 1)! σ(u (1) + u (2) + · · · + u (n) ) i<j σ(u (i) − u (j) ) σ(u (1) ) n σ(u (2) ) n · · · σ(u (n) ) n = 1 ℘(u (1) ) ℘ (u (1) ) ℘ (u (1) ) · · · ℘ (n−2) (u (1) ) 1 ℘(u (2) ) ℘ (u (2) ) ℘ (u (2) ) · · · ℘ (n−2) (u (2) …”

Section: Introductionunclassified

“…with an appropriate choice of path of the integral. Then (1.1) and (1.2) are easily rewritten as σ(u (1) + u (2) …”

Section: Introductionmentioning

confidence: 99%

“…σ(u (1) ) n σ(u (2) ) n · · · σ(u (n) ) n = 1 x(u (1) ) y(u (1) ) x 2 (u (1) ) yx(u (1) ) x 3 (u (1) ) · · · 1 x(u (2) ) y(u (2) ) x 2 (u (2) ) yx(u (2) ) x 3 (u (2) The author recently gave a generalization of the formulae (1.3) and (1.4) to the case of genus two in [19] and to the case of genus three in [20]. The aim of this paper is to generalize (1.3), (1.4) and the results in [19,20] to all hyperelliptic curves (see Theorems 8.2 and 9.3).…”

Section: Introductionmentioning

confidence: 99%

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Determinant Expressions for Hyperelliptic Functions

Ônishi¹

2005

Proceedings of the Edinburgh Mathematical Society

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“…k (u g ) the kth complete symmetric function, namely the sum of all monomials of total degree k of the variables u (1) g , . .…”

Section: The Schur-weierstrass Polynomialmentioning

confidence: 99%

Section: This Polynomial S(u) Is Called the Schur-weierstrass Polynommentioning

confidence: 99%

Section: Introductionunclassified

“…with an appropriate choice of path of the integral. Then (1.1) and (1.2) are easily rewritten as σ(u (1) + u (2) …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Determinant Expressions for Hyperelliptic Functions

Ônishi¹

2005

Proceedings of the Edinburgh Mathematical Society

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References

2022

Integrable Systems

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On the Failure Locus of Higher Order Properties of Embeddings in Projective Spaces

Ballico

1993

Mathematische Nachrichten

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IntroductionIn [BFS] the authors introduced several notions for embeddings of smooth surfaces (or of general complete varieties) in a projective space which capture some of the higher order properties of the embeddings and whose study is possible using adjunction theoretical methods: the notions of k-spannedness, k-very ampleness and of jet ampleness. We recall (very roughly) only the first and the second one. Fix an integer k 2 0. Let X be a complete variety and i a rational map from X to a projective space n. Recall that a 0-dimensional scheme 2 c X is said to be curvilinear if for every x E Zred. 2 has embedding dimension at most 1 at x; i is called 0-spanned if it is a morphism; if k > 0, i is called k-spanned (resp. k-very ample) if it is (k -1)-spanned and for every curvilinear subscheme 2 c X with length (2) = k + 1 (resp. for every subscheme 2 c X with length (2) I k + l), the linear span (i(2)) of i(2) in I 7 has dimension k. For smooth X,i is 1-spanned if and only if it is an embedding (and if and only if it is 1-very ample). A line bundle L on X is said to be k-spanned (resp. k-very ample) if the rational map induced by H o ( X , L) is k-spanned (resp. k-very ample). Here we consider similar notions not on all X , but (essentially) on a Zariski open subset. We say that ( X , W ) (or ( X , L ) ) is generically k-spanned if for a general P E X and every curvilinear subscheme 2 of X with length (2) = k + 1 and P E Zred the restriction map from W to Ho(Z, L I 2) is surjective; this definition is due to SOMMESE. We say that ( X , W ) (or ( X , L ) ) is almost everywhere k-spanned if for a general P E X and every curvilinear subscheme 2 of X with length(2) I (k + 1) and P E Zred either the restriction map rZ;+,: W + Ho(Z, L 12) is surjective or the restriction map rZ,;+,: W + Ho(Z', LI Z ) is not surjective, where Z' is the union of the connected components of 2 not containing P. The reason to introduce the notion of almost everywhere k-spannedness is that if ( X , W ) is not (k -1)-spanned, then ( X , W ) cannot be generically k-spanned: if Z' is a length k cycle for which the restriction map W + Ho(Z', L 12') is not surjective, taking Z:=Z'U {PI with P general shows this; essentially ( X , W ) is almost everywhere k-spanned if and only if the union of minimal bad 0-dimensional cycles (for W ) of length at most k + 1 is contained in proper sub-

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Geometry of Algebraic Curves

Cited by 1,498 publications

References 0 publications

Determinant Expressions for Hyperelliptic Functions

Determinant Expressions for Hyperelliptic Functions

References

On the Failure Locus of Higher Order Properties of Embeddings in Projective Spaces

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