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BIRS Workshop Lecture Videos

Negative curves on ${\bf P}^2$ Harbourne, Brian


Consider a plane curve of degree d with points $p_1,\ldots,p_s$ of multiplicities $m_1=mult_{p_1}(C),\ldots,m_s=mult_{p_s}(C)$. Define $Q(C,p_1,\ldots,p_s)$ to be $d^2-\sum_im_i^2$ and say that $C$ is negative curve of $Q<0$. A well known conjecture states that there are no integral plane curves $C$ with $Q(C,p_1,\ldots,p_s)<-1$ for generic points $p_i$. More generally, suppose we have points $q_1,\ldots,q_r$ in addition to the generic points $p_i$. Can there be integral negative curves $C$ with $Q(C,p_1,\ldots,p_s,q_1,\ldots,q_r)<-1$ if $mult_{p_i}(C)>0$ for some $i$ If so, what can be said about the occurrence of such curves Work of R. De Gennaro, D. Faenzi, G. Ilardi and J. Valles shows in general it can happen so the problem now is to classify such occurrences. I'll discuss some joint work with D. Cook II, J. Migliore and U. Nagel giving partial results.

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