Symbolic powers of codimension two Cohen-Macaulay ideals Academic Article uri icon

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abstract

  • Let $I_X$ be the saturated homogeneous ideal defining a 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 known about this problem when $X$ is locally a complete intersection, and in particular, we review the classification of when $I_X^{(m)} = I_X^m$ 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 known results about symbolic powers of ideals of points in $\mathbb{P}^1 \times \mathbb{P}^1$; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of R\"omer.

authors

  • Van Tuyl, Adam
  • Cooper, Susan
  • Fatabbi, Giuliana
  • Guardo, Elena
  • Lorenzini, Anna
  • Migliore, Juan
  • Nagel, Uwe
  • Seceleanu, Alexandra
  • Szpond, Justyna
  • Tuyl, Adam Van