On the degree of curves vanishing at fat points with equal multiplicities (Q2846870)

The problem of determining the least degree of plane curves vanishing at given points with certain multiplicities is fascinating and difficult. In 1959, Nagata gave a conjecture on the lower bound of the degree. Since then, many mathematicians have focused on solving the conjecture for several special cases. In this paper, the author show several results on determining the least degree for certain cases of equal multiplicities.NEWLINENEWLINEConsider a projective plane over an algebraically closed field with characteristic zero. For positive integers \(n\) and \(m_1,\dots,m_n\) , define \(\delta(m_1,\dots,m_n)\) to be the least integer \(d\) such that for general points \(p_1,\dots,p_n\) on \(\mathbb{P}^2\), there is a plane curve of degree \(d\) vanishing at each point \(p_i\) with multiplicity at least \(m_i\), respectively. When \(m_1=\dots=m_n=m\), this number is denoted by \(\delta(n;m)\). For \(n>9\), Nagata conjectured that \(\delta(m_1,\dots,m_n)>(1/{\sqrt{n}})\sum_{i=1}^{n}m_i.\)NEWLINENEWLINEUp to now, the conjecture has been proved only for particular cases. In this paper, the authors use combinatorial methods to obtain the following lower bounds.NEWLINENEWLINE1) Let \(m,k\) be positive integers. Then NEWLINE\[NEWLINE\delta(m^2+(2k-1)m+k^2-2k+2;m)\geq m^2+kmNEWLINE\]NEWLINENEWLINENEWLINE2) Let \(m,k,a\) be positive integers with \(0<a<m\). Then NEWLINE\[NEWLINE\delta(m^2+(2k-1)m+k^2-2k+2+q;m)\geq m^2+km+a,NEWLINE\]NEWLINE where \(q\) is a suitable integer.NEWLINENEWLINE3) Let \(m,s,a\) be positive integers with \(0<a<m\). Then NEWLINE\[NEWLINE\delta(s^2+q;m)\geq sm+a,NEWLINE\]NEWLINE where \(q\) is a suitable integer.NEWLINENEWLINEFrom the above lower bounds the authors show exact values for several cases (see the last section).