Are symbolic powers highly evolved? (Q2871001)

In this paper the authors search for structural reasons behind old results and conjectures of Chudnovksy regarding the least degree of a nonzero form in an ideal of fat points in \(\mathbb{P}^N\) and make conjectures which explain them. These conjectures were also partly motivated by the Eisenbud-Mazur Conjecture on evolutions, which concerns symbolic squares of prime ideals in local rings:NEWLINENEWLINEConjecture. Let \(P \subset \mathbb{C}[[x_1,\dots,x_d]]\) be a prime ideal. Then \(P^{(2)} \subseteq MP\), where \(M = (x_1,\dots,x_d)\). \vskip0.3cm In contrast with the previous conjecture the authors consider higher symbolic powers of homogeneous ideals in polynomial rings.NEWLINENEWLINEThe paper is strictly related to the concept of fat points. As a matter of fact, in the case of defining ideals of points, symbolic powers are special cases of ideals of fat points, and their study provides a meeting ground of geometry and algebra.NEWLINENEWLINEThe symbolic power of an ideal \(I\) is \(I^{(m)}=R\cap(\cap_P(I^m)_P)\) where the intersections take place in the field of fractions of \(K[\mathbb{P}^N]\), and the second intersection is over all associated primes \(P\) of \(I\).NEWLINENEWLINEIf \(I\) is a homogeneous ideal, we let \(\alpha(I)\) be the least degree of a polynomial in \(I\). Let \(I\) be the radical ideal of a finite set of points in \(\mathbb{P}^N\). Using complex analytic techniques, Waldschmidt and Skoda showed that NEWLINE\[NEWLINE\alpha(I^{(m)})/m \geq \alpha(I)/N NEWLINE\]NEWLINE for every \(m>0\). When \(N=2\), Chudnovsky improved the bound of Waldschmidt and Skoda. Since only a sketch of Chudnovksy's proof is given in his paper, the authors give a new proof in this paper.NEWLINENEWLINEChudnovsky's improvement is the following. Let \(p_1,\dots, p_n\in \mathbb{P}^2\) be distinct points and let \(I = \bigcap_iI(p_i)\subset K[\mathbb{P}^2]\). Then NEWLINE\[NEWLINE\alpha(I^{(m)})/m \geq (\alpha(I)+1)/2.NEWLINE\]NEWLINE This led us to the first conjecture of the authors, which gives a structural reason for the result of Chudnovsky.NEWLINENEWLINEConjecture. Let \(I=\bigcap_iI(p_i)^{m_i} \subset K[\mathbb{P}^n]\) be any fat points ideal. Then \(I^{(rN)} \subset M^{r(N-1)}I^r\) holds for all \(r>0\). \vskip0.3cm The authors prove the conjecture for the cases of general points in \(\mathbb{P}^2\) and stars configurations of points in \(\mathbb{P}^N\).NEWLINENEWLINEIn the last section of the papers the authors introduce new conjectures.NEWLINENEWLINEConjecture (4.1.1). Let \(I\subset K[\mathbb{P}^N]\) be a homogeneous ideal. Then \(I^{(rN-(N-1))} \subset I^r\) holds for all \(r\).NEWLINENEWLINEConjecture (4.1.4). Let \(I \subset K[\mathbb{P}^2]\) be the homogeneous radical ideal of a finite set of points. Then \(I^{(m)} \subset I^r\) holds whenever \(m/r \geq 2\alpha(I)/(\alpha(I) + 1)\).NEWLINENEWLINEConjecture (4.1.5). Let \(I\subset K[\mathbb{P}^N]\) be the ideal of a finite set of points \(p_i \in \mathbb{P}^N\). Then \(I^{(rN-(N-1))}\subset M^{(r-1)(N-1)}I^r\) holds for all \(r\geq 1\).NEWLINENEWLINEConjecture (4.1.8). Let \(I\subset K[\mathbb{P}^N]\) be the ideal of a finite set of points \(p_i \in \mathbb{P}^N\). Then NEWLINE\[NEWLINE\alpha(I^{(rN-(N-1))}) \geq r\alpha(I) + (r-1)(N - 1)NEWLINE\]NEWLINE for every \(r > 0\).NEWLINENEWLINEMoreover the authors prove additional results based on these conjectures.