Chapter 11 Symbolic Defect In this chapter we introduce the symbolic defect of a homogeneous ideal. This concept was introduced recently by Galetto, Geramita, Shin, and Van Tuyl [79]. There are a number of interesting questions one can ask about this invariant, and hopefully this chapter will inspire you to investigate the symbolic defect of your favourite family of homogeneous ideals. Throughout this lecture, we will assume that R = K[x 1 ,...,x n ] is a polynomial ring over an algebraically closed field of characteristic zero, and I will be a homogeneous ideal of R. 11.1 Introducing the Symbolic Defect We begin with some observations. For any homogeneous ideal I , we always have I m ⊆ I (m) . As a consequence the R-module I (m) /I m is well-defined. The main idea behind the symbolic defect of an ideal is that I (m) /I m is somehow a measure of the “failure” of I m to equal I (m) . That is, the “bigger” the module I (m) /I m , the more I m fails to equal I (m) . This suggests we may wish to study the module I (m) /I m in more detail. Interestingly, there are only a few papers on this module; the papers of which we know include papers by Arsie and Vatne [4], Herzog [104], Herzog and Ulrich [108], Huneke [116], Schenzel [152], and Vasconcelos [165]. But what do we mean by “bigger”? Note that when I is a homogeneous ideal, the R-module I (m) /I m is also a graded R-module (and also an R/I m - module). Furthermore, since R is Noetherian, the module I (m) /I m is Noetherian. Consequently, the quotient I (m) /I m is a finitely generated graded R-module, and furthermore, the number of minimal generators is an invariant of I (m) /I m . So, one way to measure “bigger” is determine the number of minimal generators of I (m) /I m . For any R-module M, let μ(M) denote the number of minimal generators of M. We can then define the symbolic defect of an ideal. © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 E. Carlini et al., Ideals of Powers and Powers of Ideals, Lecture Notes of the Unione Matematica Italiana 27, https://doi.org/10.1007/978-3-030-45247-6_11 89