Let $I_X$ be the saturated homogeneous ideal defininga codimension two arithmetically Cohen-Macaulay scheme $X subseteq mathbb{P}^n$, and let $I_X^{(m)}$ denote its $m$-th symbolic power.We are interested in when $I_X^{(m)} = I_X^m$. We survey what is knowabout this problem when $X$ is locally a complete intersection,and in particular, we review the classification ofwhen $I_X^{(m)} = I_X^m$ for all $m geq 1$. We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally,we show that this classification allows one to: (1) simplify knownresults about symbolic powers of ideals of points in $popo$; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of R"omer.
Symbolic powers of codimension two Cohen-Macaulay ideals
Fatabbi, Giuliana;
2020
Abstract
Let $I_X$ be the saturated homogeneous ideal defininga codimension two arithmetically Cohen-Macaulay scheme $X subseteq mathbb{P}^n$, and let $I_X^{(m)}$ denote its $m$-th symbolic power.We are interested in when $I_X^{(m)} = I_X^m$. We survey what is knowabout this problem when $X$ is locally a complete intersection,and in particular, we review the classification ofwhen $I_X^{(m)} = I_X^m$ for all $m geq 1$. We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally,we show that this classification allows one to: (1) simplify knownresults about symbolic powers of ideals of points in $popo$; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of R"omer.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
