Alexander–Hirschowitz Theorem

• The Alexander–Hirschowitz theorem is a foundational result in algebraic geometry that determines when a set of double points in projective space imposes independent conditions on homogeneous forms.
• It connects interpolation problems with the geometry of secant varieties and Veronese embeddings using tools such as Terracini’s lemma and degeneration techniques.
• The theorem’s implications extend to weighted projective spaces, smooth surfaces, and higher multiplicity cases, driving ongoing research in Hilbert functions and defective loci.

The Alexander–Hirschowitz theorem is the foundational result in the interpolation problem for double points in algebraic geometry. It describes precisely when a general collection of double points in projective space imposes independent conditions on the linear system of homogeneous forms of given degree, and classifies the exceptional cases where the naive count fails. The theorem has broad implications ranging from the study of secant and Terracini loci to the geometry of Veronese embeddings and secant varieties.

1. Classical Statement and Expected Dimension

Let $n \geq 1$ and %%%%1%%%% be integers. The space of homogeneous degree $d$ forms in $n+1$ variables is $$V_{n,d} := H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d))$$, of dimension $\binom{n+d}{d}$. Given $m$ general points $p_1, \ldots, p_m \in \mathbb{P}^n$, imposing a double point condition at $p_i$ (i.e., singularity at $p_i$) translates to

• $f(p_i) = 0$,
• All first partial derivatives vanish at $p_i$.

This imposes $(n+1)$ linear conditions at each $p_i$. Thus, for $m$ points, one expects the space of degree $d$ hypersurfaces singular at all $p_i$ to have dimension

$$\operatorname{expdim} = \max\left\{-1, \binom{n+d}{d} - m(n+1) - 1 \right\}.$$

The Alexander–Hirschowitz theorem states that for $n$, $d$, $m$ as above, the expected count is always correct except for a specific list of "defective" cases:

• $d=2$ and $2 \leq m \leq n$,
• $(n, d, m) = (2,4,5),\ (3,4,9),\ (4,3,7),\ (4,4,14)$ (Lian, 2020, Ha et al., 2021, Postinghel, 2010, Roshan-Zamir, 2024).

In each exception, the double points fail to impose independent conditions, i.e. the imposed codimension is strictly less than $m(n+1)$. In all other parameters, the expected dimension is achieved.

2. Secant Varieties, Veronese Embedding, and Terracini’s Lemma

A central insight is that the interpolation problem for double points is equivalent to the geometry of higher secant varieties to the Veronese variety. The degree $d$ Veronese embedding $v_d : \mathbb{P}^n \hookrightarrow \mathbb{P}^{N}$ (with $N = \binom{n+d}{d} - 1$) sends $p \mapsto$ all degree $d$ monomials evaluated at $p$.

The $s$–th secant variety $\sigma_s(V_{n,d})$ is the closure of the union of $(s-1)$–planes spanned by $s$ points on $V_{n,d}$. The classical expectation is

$$\dim \sigma_s(V_{n,d}) = \min\{ s(n+1)-1, \binom{n+d}{d}-1 \}.$$

Terracini’s lemma states that, for $s$ general points $p_1,\ldots,p_s \in V_{n,d}$, the tangent spaces at $p_i$ span the tangent space to the secant variety at a general point of their linear span. The defectivity (i.e., when the actual dimension is smaller) matches precisely the exceptional cases of the Alexander–Hirschowitz theorem (Ha et al., 2021, Roshan-Zamir, 2024).

3. Proof Methods: Induction, Degeneration, and the Horace Differential

The original proof by Alexander and Hirschowitz uses a refined induction on $(n,d)$, with tools including:

• Degeneration: Specializing points onto hyperplanes and analyzing the resulting schemes (“Horace method”).
• Differential Horace: Further specialization via 1-parameter families, using “trace” and “residue” arguments, and semi-continuity of the Hilbert function.
• Castelnuovo’s inequality: To relate Hilbert functions in degree $d$ to those in smaller degrees.
• Explicit computation and computer verification: For the finite collection of low-degree, small-$n$ exceptional cases.

Degeneration approaches have since been streamlined: for example, (Postinghel, 2010) introduces a uniform degeneration strategy via blow-ups to identify the exceptional locus, while (Ha et al., 2021) and (Roshan-Zamir, 2024) clarify the connection to secant varieties and tangent space calculations via Terracini (Lian, 2020).

4. Defective Cases, Terracini Loci, and Codimension Phenomena

The full-dimensional Terracini locus $\mathbb{T}(n, d; x)$ parametrizes sets $S \subset \mathbb{P}^n$ of $x$ points spanning $\mathbb{P}^n$ for which the double-point interpolation fails. The Alexander–Hirschowitz theorem classifies $(n,d,x)$ where $\dim \mathbb{T}(n,d;x) = xn$.

Refinements, such as those in (Ballico et al., 2024), go further: for $n=2$ (planar case), the locus where the dimension first drops by one is characterized completely. The possible $(d,x)$ where $\mathbb{T}(2,d;x)$ has a component of dimension $2x-1$ are:

• $(4,4)$,
• $(6,10)$,
• $d \equiv 1,2 \pmod{3}$, $d \geq 7$, $x = (d+2)(d+1)/6$.

The paper constructs these loci via Severi varieties of nodal plane curves and via explicit combinatorial conditions on point configurations (e.g., points lying partially on a single line). Outside these, cohomological dimension bounds force lower-dimensional loci (Ballico et al., 2024).

5. Generalizations: Higher Multiplicity, Weighted Projective Space, and Surfaces

Extensions of the theorem address:

• Weighted projective space: The interpolation theory and the Alexander–Hirschowitz paradigm carry over for well-formed spaces with at least one weight one. The expected dimension formula applies modulo analogous defects, which can be analyzed by lattice-theoretic (monomial) arguments, and adaptations of Terracini's lemma (Roshan-Zamir, 2024).
• Smooth projective surfaces: For a smooth complex projective surface $X$ and ample $L$, (Lian, 2020) proves that for sufficiently large $d$, all collections of double points on $X$ impose independent conditions on $|L^d|$; this is an “asymptotic Alexander–Hirschowitz theorem for surfaces.” The proof leverages vanishing theorems (Kawamata–Viehweg, Serre) and comparisons of quadratic versus linear growth rates in $h^0(X,L^d)$ and $h^0(X,M)$, with $M$ arising from the geometry of base loci.

A conjectural extension is that for smooth projective varieties of any dimension and ample line bundle, the independence of general double points is eventually achieved for all large $d$ (Lian, 2020).

6. Open Problems and Future Directions

Current open questions include:

• Classification of Hilbert functions for arbitrary double-point schemes in $\mathbb{P}^n$ (not necessarily general configurations).
• Minimal and maximal Hilbert functions in the space of double-point Hilbert functions.
• Classification of defective cases for higher multiplicities ($m\geq 3$): Notably, Nagata’s conjecture and the SHGH conjecture remain unresolved.
• Symbolic versus ordinary powers: Measuring Hilbert function defects and counting extra generators in $I^{(2)}$ versus $I^2$.
• Interpolation on special subvarieties, e.g., rational normal curves.
• Higher-dimensional and arithmetic generalizations: Sharpening intersection-theoretic bounds to fully generalize the asymptotic theorem beyond surfaces.
• Extensions to multiprojective and weighted projective spaces: Some cases are settled, but further classification is ongoing (Ha et al., 2021, Roshan-Zamir, 2024, Lian, 2020).

The Alexander–Hirschowitz theorem thus serves as a central result in the interplay between interpolation theory, secant varieties, and the geometry of high-degree subvarieties in projective and other ambient spaces.