Base Ideal of Complete Linear System

• The base ideal of a complete linear system is the scheme-theoretic locus where all sections of a line bundle vanish, dictating fixed components and singularity behavior.
• It plays a crucial role in controlling invariants such as log canonical thresholds and regularity, particularly in contexts like plane Cremona maps and abelian varieties.
• Homological methods, vanishing theorems, and Fourier–Mukai transforms are key tools for understanding its structure and broader implications in algebraic geometry.

The base ideal of a complete linear system encodes the scheme-theoretic locus where all sections of a given line bundle on a projective variety simultaneously vanish. This ideal plays a central role in the birational and homological study of linear systems, determining the fixed part of the system, controlling its singularities, and influencing both geometric and homological invariants. The structure and singularities of base ideals are especially significant in the study of abelian varieties and rational maps such as plane Cremona transformations, where they govern vanishing theorems, canonical thresholds, regularity, and the behavior of associated maps.

1. Definition and Basic Properties

Let $X$ be a complex projective variety and $L$ an effective line bundle on $X$. The complete linear system %%%%3%%%% is the projective space $\mathbb P(H^0(X, L))$; its base locus $\operatorname{Bs}|L| \subset X$ is the (scheme-theoretic) common zero locus of all sections of $L$. The base ideal sheaf is defined by

$$I(|L|) = \mathrm{Im}\left( H^0(X, L) \otimes_\mathbb C L^{-1} \xrightarrow{\mathrm{ev}} \mathcal O_X \right) \subset \mathcal O_X,$$

so that $I(|L|)\cdot L \subset \mathcal O_X$ is the ideal sheaf of $\operatorname{Bs}|L|$ (viewed inside $L$) (Pareschi, 21 Dec 2025). In the case of plane curves over an algebraically closed field $k$, let $R = k[x, y, z]$ and $V \subset R_d$ be a three-dimensional vector space with a basis $(f_0, f_1, f_2)$, which defines

$I = (f_0, f_1, f_2) \subset R.$

Here, $I$ is the (homogeneous) base ideal, whose zero-scheme is the indeterminacy locus of the associated rational map (Hassanzadeh et al., 2011).

2. Base Ideal and Log Canonical Thresholds

For a smooth variety $X$ and a nonzero coherent ideal sheaf $\mathcal I \subset \mathcal O_X$, the log canonical threshold (lct) is defined as

$$\operatorname{lct}(\mathcal I) = \inf \left\{ c > 0 : (X, c \cdot Z(\mathcal I)) \text{ is not log canonical} \right\} = \inf \{ c > 0 : \mathcal J(\mathcal I^c)\neq \mathcal O_X \},$$

where $\mathcal J(\mathcal I^c)$ denotes the multiplier ideal of $\mathcal I$ to exponent $c$ (Pareschi, 21 Dec 2025).

Let $A$ be a complex abelian variety of dimension $g$ and let $L$ be any ample line bundle. Writing $b_L = I(|L|)$ for the base ideal, the principal result is:

• For every ample $L$ on $A$, $\operatorname{lct}(b_L) \geq 1$.
• Equality ($\operatorname{lct}(b_L) = 1$) holds if and only if $\operatorname{Bs}|L|$ contains a divisorial component.
• If $|L|$ is base-point-free up to codimension two (i.e., base scheme without divisorial part), then $\operatorname{lct}(b_L) > 1$ (Pareschi, 21 Dec 2025).

This establishes a dichotomy: the singularities of the base scheme are always at least log canonical, with non-trivial fixed divisors exactly characterizing the boundary case.

3. Homological Structure and Resolution of Base Ideals

In the context of the plane ($\mathbb P^2$), when $V \subset R_d$ defines a net of degree $d$ plane curves, the base ideal $I = (f_0, f_1, f_2)$ is of codimension 2. Its homological properties, including saturation and regularity, are tightly constrained for linear systems associated to Cremona maps. The following results hold (Hassanzadeh et al., 2011):

• For any base ideal $I$, its saturation and the quotient $I^{\text{sat}} / I$ satisfy a self-duality, with

$$\operatorname{indeg}(I^{\text{sat}} / I) + \operatorname{end}(I^{\text{sat}} / I) = 3d - 3.$$

• For Cremona maps, $\operatorname{indeg}(I^{\text{sat}} / I) \ge d + 1$.
• The regularity of $R / I$,

$\operatorname{reg}(R / I) \le 2d - 3,$

with explicit bounds and resolution formats for degrees 5, 6, and 7.

• The minimal graded free resolution of $I$ is determined (for height 2 ideals generated by 3 forms of degree $d$) and the "homaloidal type" summarizes the sequence of multiplicities and their algebraic constraints, e.g.,

$$\sum \mu_i = 3d - 3, \qquad \sum \mu_i^2 = d^2 - 1.$$

4. Companions and Inclusions: Saturation, Integral Closure, and Other Ideals

The base ideal $I$ participates in a natural ladder of homogeneous ideals sharing the same radical, reflecting various geometric and valuative constraints: $$I \subset \overline{I} \subset \tilde{I} \subset I_K \subset \mathfrak K^{\text{fat}} \subset I^{\text{sat}},$$ where:

• $\overline{I}$ is the integral closure,
• $\tilde{I}$ is the divisorial cover,
• $I_K$ and $\mathfrak K^{\text{fat}}$ are companion ideals corresponding to the weighted base cluster,
• $I^{\text{sat}}$ is the saturation with respect to the irrelevant ideal (Hassanzadeh et al., 2011).

For Cremona maps, this chain often collapses: $$\mathfrak K^{\text{fat}} = I^{\text{sat}} = \overline{I}$$ (integrally closed base locus).

5. Fourier–Mukai Transforms and Generic Vanishing

For abelian varieties, the structure of the base ideal and its multiplier ideal is intimately connected with generic vanishing theory, functorial properties of the Fourier–Mukai transform, and invariance under natural theta-group actions:

• Nadel vanishing implies vanishing of all higher cohomologies for $L \otimes \mathcal J(b_L^c)$ for $i > 0$ and $0 < c < 1$.
• The symmetric Fourier–Mukai transform applied to $F = L \otimes \mathcal J(b_L^c)$ yields a locally free sheaf; the non-freeness (vanishing) criterion induces constraints on the singularity of the base locus.
• The invariance under the theta group $K(L)$ forces a dichotomy for global sections, facilitating detection of divisorial fixed components (Pareschi, 21 Dec 2025).

6. Examples, Corollaries, and Applications

Table: Consequences and Examples

Property	Base Ideal Condition	Consequence
$|L|$ very ample or base-point-free in codim 1	No fixed divisor	$\operatorname{lct}(b_L) > 1$
$\operatorname{Bs}|L|$ contains prime divisor $D$	$b_L = \mathcal O_A(-D)$ locally	$\operatorname{lct}(b_L) = 1$
Plane cubic net	Saturated base ideal	Hilbert–Burch resolution, regularity $2$

A notable corollary is the support for Debarre’s conjecture: the lower bound on $\operatorname{lct}(b_L)$ suggests that all codimension 2 components of the base scheme are generically reduced, although this is not proven (Pareschi, 21 Dec 2025).

For plane Cremona maps:

• For degree $d \leq 4$, all base ideals are saturated.
• When not saturated, explicit free resolutions are determined, with unique homaloidal types for non-saturated cases in degrees 5, 6, and 7 (Hassanzadeh et al., 2011).

7. Extensions and Significance

The study of base ideals of complete linear systems bridges classical questions of projective geometry—including the birational geometry of abelian varieties, the structure of plane mapping classes, and regularity bounds for algebraic invariants—with advanced techniques from multiplier ideals, generic vanishing, and Fourier–Mukai theory. The results generalize classical theorems on the singularity of theta divisors (Kollár, Ein–Lazarsfeld), displaying a clear dichotomy: the base ideal distinguishes linear systems with divisorial fixed loci from those with only mild (log canonical or better) singularities (Pareschi, 21 Dec 2025). This dichotomy continues to inform the study of higher-dimensional birational geometry, vanishing theorems, and the algebraic structure of companion ideals. The geometric, algebraic, and categorical facets of the base ideal remain an active domain for further research and refinement.