Upper bound on the number of generators of a local complete intersection curve in $\mathbb{P}^3$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T13:03:06Z http://mathoverflow.net/feeds/question/112301 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112301/upper-bound-on-the-number-of-generators-of-a-local-complete-intersection-curve-in Upper bound on the number of generators of a local complete intersection curve in $\mathbb{P}^3$ Naga Venkata 2012-11-13T17:02:45Z 2012-11-28T18:22:00Z <p>Let $C$ be a local complete intersection curve in $\mathbb{P}^3$ (not irreducible or smooth) of degree $e$. Suppose $f_1, f_2$ (and $e_i=\deg(f_i)$) are two of the lowest degree generators of $I(C)$. Then:</p> <p>1) Is it true that the minimal number of generators of $I(C)$ is less than or equal to $2+(e_1e_2-e)$? (Intuitively I would think this to be true by using the degree formula from intersection theory which says that $e_1e_2=\sum_im_i\deg(C_i)$ where $C_i$ ranges over the irreducible components of the curve defined by $f_1$ and $f_2$ and $m_i$ are their multiplicities)</p> <p>2) Is there any known result/approach to compute the upper bound on the minimal number of generators of $I(C)$ in terms of its degree?</p> http://mathoverflow.net/questions/112301/upper-bound-on-the-number-of-generators-of-a-local-complete-intersection-curve-in/112398#112398 Answer by Nikita Kalinin for Upper bound on the number of generators of a local complete intersection curve in $\mathbb{P}^3$ Nikita Kalinin 2012-11-14T17:54:27Z 2012-11-14T17:54:27Z <p>"every locally complete intersection curve in P3 can be defined by four equations."</p> <p>look at <a href="http://www.math.binghamton.edu/somnath/Notes/curves.pdf" rel="nofollow">http://www.math.binghamton.edu/somnath/Notes/curves.pdf</a> and <a href="http://www.math.tifr.res.in/~publ/ln/tifr62.pdf" rel="nofollow">http://www.math.tifr.res.in/~publ/ln/tifr62.pdf</a></p>