[Paper Review] Symbolic powers of ideals of generic points in P^3

Marcin Dumnicki|arXiv (Cornell University)|May 2, 2011

Commutative Algebra and Its Applications5 references4 citations

TL;DR

This paper proves the Harbourne–Huneke conjecture on symbolic powers for ideals of generic points in $\mathbb{P}^3$ and for at most $N+1$ generic points in $\mathbb{P}^N$, showing $I^{(Nr)} \subset M^{(N-1)r}I^r$ for all $r \geq 1$. As a consequence, Chudnovsky's conjecture holds for generic points in $\mathbb{P}^3$, with explicit lower bounds on the Waldschmidt constant $\gamma(I)$. The proof uses Cremona transformations and monomial ideal techniques on fundamental points.

ABSTRACT

B. Harbourne and C. Huneke conjectured that for any ideal $I$ of fat points in $P^N$ its $r$-th symbolic power $I^{(r)}$ should be contained in $M^{(N-1)r}I^r$, where $M$ denotes the homogeneous maximal ideal in the ring of coordinates of $P^N$. We show that this conjecture holds for the ideal of any number of simple (not fat) points in general position in $P^3$ and for at most $N+1$ simple points in general position in $P^N$. As a corollary we give a positive answer to Chudnovsky Conjecture in the case of generic points in $P^3$.

Motivation & Objective

• To verify the Harbourne–Huneke conjecture for ideals of generic points in $\mathbb{P}^3$ and for at most $N+1$ generic points in $\mathbb{P}^N$.
• To establish containment $I^{(Nr)} \subset M^{(N-1)r}I^r$ for symbolic powers of ideals of generic points.
• To provide a positive answer to Chudnovsky's conjecture for generic points in $\mathbb{P}^3$.
• To compute lower bounds for the Waldschmidt constant $\gamma(I)$ of ideals of generic points in $\mathbb{P}^3$.

Proposed method

• Use of Cremona transformations to relate linear systems of different degrees and multiplicities.
• Reduction of general configurations of generic points to fundamental points via linear automorphisms.
• Explicit description of symbolic powers $I^{(m)}$ as monomial ideals via degree constraints on monomials.
• Construction of factorizations $x^{(a_0,\dots,a_N)} = y \cdot z$ with $y \in I^r$ and $\deg(z) \geq (N-1)r$ to prove containment.
• Application of the limit definition of the Waldschmidt constant $\gamma(I) = \lim_{m \to \infty} \alpha(I^{(m)})/m$ to derive bounds.
• Use of monomial ideal intersection properties: $I \cap J = \text{span of monomials in } I \cap J$.

Experimental results

Research questions

• RQ1 Does the Harbourne–Huneke conjecture $I^{(Nr)} \subset M^{(N-1)r}I^r$ hold for ideals of generic points in $\mathbb{P}^3$?
• RQ2 Can the conjecture be extended to at most $N+1$ generic points in $\mathbb{P}^N$?
• RQ3 Does Chudnovsky's conjecture $\alpha(I^{(m)}) \geq m(\alpha(I) + N - 1)/N$ hold for generic points in $\mathbb{P}^3$?
• RQ4 What are effective lower bounds for the Waldschmidt constant $\gamma(I)$ of ideals of generic points in $\mathbb{P}^3$?
• RQ5 How can symbolic powers of ideals of generic points be described using monomial bases?

Key findings

• The Harbourne–Huneke conjecture holds for any number of generic points in $\mathbb{P}^3$, with $I^{(3r)} \subset M^{2r}I^r$ for all $r \geq 1$.
• The conjecture also holds for at most $N+1$ generic points in $\mathbb{P}^N$, with $I^{(Nr)} \subset M^{(N-1)r}I^r$.
• Chudnovsky's conjecture is confirmed for generic points in $\mathbb{P}^3$, with $\gamma(I) \geq \frac{s+1}{3}$ for $s$ points, $s \geq 5$.
• For $s \geq 8$ generic points in $\mathbb{P}^3$, $\gamma(I) \geq 2$; for $s=5$, $\gamma(I) \geq \frac{5}{3}$.
• The Waldschmidt constant satisfies $\gamma(I) \geq \gamma(I(1^{\times n}))$ for $n$ generic points, and $\gamma(I^{(r)}) = r\gamma(I)$.
• The symbolic power $I^{(m)}$ of an ideal of fundamental points is generated by monomials $x^{(a_0,\dots,a_N)}$ with $\sum a_j = t$ and $a_k \leq t - m$ for $k=0,\dots,n-1$.