We show that given integers $N$, $d$ and $n$ such that ${N\ge2}$,
${(N,d,n)\ne(2,2,5)}$, and ${N+1\le n\le\tbinom{d+N}{N}}$, there is a family of
$n$ monomials in $K[X_0,\ldots,X_N]$ of degree $d$ such that their syzygy
bundle is stable. Case ${N\ge3}$ was obtained independently by Coand\v{a} with
a different choice of families of monomials [Coa09].
  For ${(N,d,n)=(2,2,5)}$, there are $5$ monomials of degree~$2$ in
$K[X_0,X_1,X_2]$ such that their syzygy bundle is semistable.Comment: 16 pages, to appear in the Proceedings of the American Mathematical
  Societ

Marques, Pedro Macias

Miró-Roig, Rosa María

Proceedings of the American Mathematical Society

English

arXiv

We show that given integers 

  
    N
    N
  

, 

  
    d
    d
  

 and 

  
    n
    n
  

 such that 

  
    
      
        N
        ≥
        2
      
      ,
      (
      N
      ,
      d
      ,
      n
      )
    
    {N\ge 2}, (N,d,n)
  

 

  
    
      ≠
      (
      2
      ,
      2
      ,
      5
      )
    
    \ne (2,2,5)
  

, and 

  
    
      N
      +
      1
      ≤
      n
      ≤
      
        
          
            
              (
            
          
          
            
              d
              +
              N
            
            N
          
          
            
              )
            
          
        
      
    
    {N+1\le n\le \tbinom {d+N}{N}}
  

, there is a family of 

  
    n
    n
  

 monomials in 

  
    
      K
      
        [
        
          X
          0
        
        ,
        …
        ,
        
          X
          N
        
        ]
      
    
    K\left [X_0,\ldots ,X_N\right ]
  

 of degree 

  
    d
    d
  

 such that their syzygy bundle is stable. Case 

  
    
      N
      ≥
      3
    
    {N\ge 3}
  

 was obtained independently by Coandǎ with a different choice of families of monomials.

For 

  
    
      (
      N
      ,
      d
      ,
      n
      )
      =
      (
      2
      ,
      2
      ,
      5
      )
    
    {(N,d,n)=(2,2,5)}
  

, there are 

  
    5
    5
  

 monomials of degree 

  
    2
    2
  

 in 

  
    
      K
      
        [
        
          X
          0
        
        ,
        
          X
          1
        
        ,
        
          X
          2
        
        ]
      
    
    K\left [X_0,X_1,X_2\right ]
  

 such that their syzygy bundle is semistable.</p

Pedro Macias Marques

Rosa Miró-Roig

Crossref

Stability of syzygy bundles

We show that given integers $N$, $d$ and $n$ such that ${N\ge2}$, ${(N,d,n)\ne(2,2,5)}$, and ${N+1\le n\le\tbinom{d+N}{N}}$, there is a family of $n$ monomials in $K[X_0,\ldots,X_N]$ of degree $d$ such that their syzygy bundle is stable. Case ${N\ge3}$ was obtained independently by Coand\v{a} with a different choice of families of monomials [Coa09].
For ${(N,d,n)=(2,2,5)}$, there are $5$ monomials of degree~$2$ in $K[X_0,X_1,X_2]$ such that their syzygy bundle is semistable

Macias Marques, Pedro

Miró Roig, Rosa María

Repositório Científico da Universidade de Évora

We show that given integers $ N$, $ d$ and $ n$ such that $ {N\ge2}, (N,d,n)$ $ \ne(2,2,5)$, and $ {N+1\le n\le\tbinom{d+N}{N}}$, there is a family of $ n$ monomials in $ K\left[X_0,\ldots,X_N\right]$ of degree $ d$ such that their syzygy bundle is stable. Case $ {N\ge3}$ was obtained independently by Coanda with a different choice of families of monomials. For $ {(N,d,n)=(2,2,5)}$, there are $ 5$ monomials of degree $ 2$ in $ K\left[X_0,X_1,X_2\right]$ such that their syzygy bundle is semistable

Macias Marques, Pedro Correia Gonçalves

Miró-Roig, Rosa M. (Rosa Maria)

Diposit Digital de la Universitat de Barcelona

http://diposit.ub.edu/dspace/bitstream/2445/96592/1/589161.pdf

Stability of syzygy bundles

Abstract

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Crossref

Repositório Científico da Universidade de Évora

Diposit Digital de la Universitat de Barcelona