Subcanonical points on projective curves and triply periodic minimal surfaces in the Euclidean space

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• 作者：Francesco Bastianelli (1)
  Gian Pietro Pirola (2)
  
  1. Dipartimento di Matematica e Fisica ; Universit脿 degli Studi Roma Tre ; Largo San Leonardo Murialdo 1 ; 00146 ; Rome ; Italy
  2. Dipartimento di Matematica ; Universit脿 degli Studi di Pavia ; Via Ferrata 1 ; 27100 ; Pavia ; Italy
  
• 关键词：Subcanonical point ; Theta ; characteristic ; Weierstrass point ; Minimal surface ; 14H55 ; 14H10 ; 53A10 ; 53C42
• 刊名：Mathematische Zeitschrift
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：279
• 期：3-4
• 页码：1029-1046
• 全文大小：294 KB
• 参考文献：1. Arbarello, E, Cornalba, M, Griffiths, PA, Harris, J (1985) Geometry of Algebraic Curves, Vol. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles in Mathematical Sciences]. Springer, New York
  2. Arezzo, C, Pirola, GP (1999) On the existence of periodic minimal surfaces. J. Algebraic Geom. 8: pp. 765-785
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  18. Mumford, D (1971) Theta-characteristic on an algebraic curve. Ann. Sci. 脡cole Norm. Sup. 4: pp. 181-192
  19. Nayatani, S (1993) Morse index and Gauss map of complete minimal surfaces in Euclidean 3-space. Comment. Math. Helv. 68: pp. 511-537 CrossRef
  20. Pirola, GP (1998) The infinitesimal variation of the spin abelian differentials and periodic minimal surfaces. Commun. Anal. Geom. 6: pp. 393-426
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  22. Teixidor i Bigas, M (1987) Half-canonical series on algebraic curves. Trans. Am. Math. Soc 302: pp. 99-116 CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

A point \(p\in C\) on a smooth complex projective curve of genus \(g\ge 3\) is subcanonical if the divisor \((2g-2)p\) is canonical. The subcanonical locus \(\mathcal {G}_g\subset \mathcal {M}_{g,1}\) described by pairs \((C,p)\) as above has dimension \(2g-1\) and consists of three irreducible components. Apart from the hyperelliptic component \(\mathcal {G}_g^\mathrm{hyp }\) , the other components \(\mathcal {G}_g^\mathrm{odd }\) and \(\mathcal {G}_g^\mathrm{even }\) depend on the parity of \(h^0(C,(g-1)p)\) , and their general points satisfy \(h^0(C,(g-1)p)=1\) and \(2\) , respectively. In this paper, we study the subloci \(\mathcal {G}_g^{r}\) of pairs \((C,p)\) in \(\mathcal {G}_g\) such that \(h^0(C,(g-1)p)\ge r+1\) and \(h^0\left( C,(g-1)p\right) \equiv r+1\,(\mathrm{mod }\,2)\) . In particular, we provide a lower bound on their dimension, and we prove its sharpness for \(r\le 3\) . As an application, we further give an existence result for triply periodic minimal surfaces immersed in the 3-dimensional Euclidean space, completing a previous result of the second author.