Seshadri Constants of Ample Adjoint Divisors

• Seshadri constants quantify the local positivity of divisors like K_X + A and link the geometry of a variety with the global behavior of its adjoint line bundles.
• Explicit lower bounds for these constants ensure criteria for global generation, very ampleness, and birational properties across surfaces, threefolds, and varieties in positive characteristic.
• Proof techniques involve intersection theory, effective divisor constructions, and Frobenius methods to achieve rationality confirmations and jet separation results.

Seshadri constants of ample adjoint divisors quantify the local positivity of divisors of the form $K_X + A$, where $K_X$ is the canonical divisor on a smooth projective variety $X$, and $A$ is ample. These constants serve as a bridge between the geometry of $X$ and the behavior of adjoint line bundles, with direct implications for global generation, very ampleness, and birational properties. The study of their lower bounds, rationality, and relation to jet separation unifies techniques across characteristic, dimension, and arithmetic, with significant results for surfaces, threefolds, and varieties in positive characteristic.

1. Definitions and General Framework

For a projective variety $V$ over an algebraically closed field, $x \in V$ a closed point, and $L$ an ample Cartier or $\mathbb{Q}$-Cartier divisor, the Seshadri constant at $x$ is

$$\varepsilon(L;x) = \inf_{C \ni x} \frac{L \cdot C}{\operatorname{mult}_x C}$$

where the infimum is over all integral curves $C$ through $x$. Alternatively, via blow-up $\pi\colon \tilde{V} \to V$ at $x$ with exceptional divisor $E$,

$$\varepsilon(L;x) = \sup\{ t \geq 0 \mid \pi^*L - tE \text{ is nef} \}$$

This concept extends to multi-point Seshadri constants and further variants in positive characteristic, such as the Frobenius–Seshadri constant $\varepsilon_F(L;x)$ and its higher-order analogues, which involve Frobenius-jet separation and have similar formal properties (Mustata et al., 2012, Murayama, 2017).

2. Lower Bounds for Seshadri Constants of Adjoint Divisors

A foundational result states that for any smooth projective variety $X$ of dimension $n$, nef line bundle $L$ with $K_X+L$ ample,

$\varepsilon(K_X+L, x) \geq \frac{2}{n^2+2n+4}$

for all $x \in X$ (Bauer et al., 2010). On surfaces ($n=2$), this general bound gives $1/6$, but the optimal bound is $1/2$ thanks to adjunction-theoretic arguments: $\varepsilon(K_X+L, x) \geq \frac{1}{2}$ If $0 < \varepsilon(K_X+L, x) < 1$, then all possible values are of the form $(m-1)/m$ with $m \geq 2$. For very ample $L$, the “hyper-adjoint” case, the bound improves to $\varepsilon(K_X+L, x) \geq 1$ (Bauer et al., 2010).

Recent work has pushed these lower bounds for more ample adjoint divisors and in arbitrary characteristic. For a smooth surface $S$ and ample $A$,

$\varepsilon(K_S+4A; x) \geq \frac{3}{4}$

for any $x \in S$ (Rösler, 22 Jan 2026). For ample $A$ with $A^2>1$, one obtains $\varepsilon(K_S+4A; x) \geq 1$. The proofs blend intersection-theoretic inequalities, adjunction, and the construction of effective $\mathbb{Q}$-divisors with prescribed vanishing order at $x$.

On threefolds, for $X$ smooth, $\varepsilon(K_X+6A; x)$ is bounded below by $1/(2\sqrt{2} - \delta)$ for any $\delta > 0$ and all but finitely many curves $C$ through $x$ (Rösler, 22 Jan 2026).

3. Rationality and Attainment of Seshadri Constants

Whenever $\varepsilon(K_X+6A; x) < 1/(2\sqrt{2})$ on a threefold, the constant is always a rational number, actually achieved by some curve $C$ (a Seshadri curve) through $x$. The proof involves combinatorial control via effective divisors with large local multiplicity and bend-and-break arguments for the exceptional set of curves. This rationality result mirrors known finiteness and rationality phenomena for Seshadri constants on abelian and bielliptic surfaces: if $\varepsilon(L, x) < \sqrt{L^2}$ and is not computed by an elliptic curve, then it is among finitely many possible rational values $d/m$ determined by intersection theory and genus constraints (Bauer et al., 2020).

4. Seshadri Constants and Jet Separation for Adjoint Bundles

A key application of lower bounds is to the separation of jets and local/global generation of adjoint line bundles. For $L$ ample on a smooth projective variety $X$ of dimension $n$, the classical Demailly criterion states that if

$\varepsilon(L;x) > n + \ell$

then $K_X \otimes L$ separates $\ell$-jets at $x$.

In positive characteristic, the Frobenius–Seshadri constant $\varepsilon_F^{(\ell)}(L;x)$ plays the analogous role: if

$\varepsilon_F^{(\ell)}(L;x) > \ell + 1$

then $K_X \otimes L$ separates $\ell$-jets at $x$ (Murayama, 2017). For $\ell=0$, the threshold is $\varepsilon_F(L;x) > 1$ for global generation, and $\varepsilon_F(L;x) > 2$ for very ampleness of $\omega_X \otimes L$ (Mustata et al., 2012). It is established that $\varepsilon_F(L;x) \geq \varepsilon(L;x)/n$, and in many cases these inequalities are sharp. In particular, thresholds for global generation and birationality of adjoint bundles in positive characteristic match those from characteristic zero, with Frobenius-trace techniques replacing vanishing theorems.

5. Detailed Results for Special Classes—Abelian and Bielliptic Surfaces

On abelian or bielliptic surfaces, where $K_X \equiv 0$, the Seshadri constant for an ample $L$ is determined completely by

$$\varepsilon(L,x) = \inf_{C \ni x} \frac{L \cdot C}{\operatorname{mult}_x C}$$

excluding the case where the minimum is achieved by an elliptic curve or Albanese fiber. Explicit lower bounds in terms of $N = L^2$ are given by

$$\varepsilon(L, x) \geq \min_{2 \leq m \leq 7} \frac{ \lceil \sqrt{ N(2 + m(m-1)) } \rceil }{ m }$$

For $N > 4982$, this minimum is achieved at $m=4$, yielding the asymptotic bound $$\varepsilon(L,x) > \frac{ \sqrt{14} }{ 4 } \sqrt{N }$$. All possible values below $\sqrt{N}$ are rational numbers $d/m$ lying in a finite list indexed by degree and multiplicity, as constrained by the Hodge index theorem and curve singularity inequalities (Bauer et al., 2020).

6. Extensions to Positive Characteristic and Frobenius Techniques

In positive characteristic, Frobenius–Seshadri constants $\varepsilon_F(L;x)$ are defined via the separation of $p^e$-Frobenius jets. The main properties are:

• $\varepsilon_F(L;x) \geq \varepsilon(L;x)/n$ and $\varepsilon_F(L;x) \leq \varepsilon(L;x)$.
• $\varepsilon_F(L;x) > 1$ (resp. $>2$) imply $\omega_X \otimes L$ is globally generated (resp. very ample) at $x$ (Mustata et al., 2012).
• Criteria for jet-separation via Frobenius–Seshadri constants match exactly the thresholds from characteristic zero, supporting a unified theory across characteristics.
• On projective space with $L = \mathcal{O}(1)$ in $\dim X = n$, $\varepsilon_F(\mathcal{O}(1); x) = 1/n$, showing that positive characteristic effects can weaken jet separation compared to the classical case.

The theory supports characterizations of projective space in characteristic $p>0$: if $X$ is a smooth Fano variety of $\dim X = n$ with $\varepsilon_F(-K_X; x) \geq n+1$ at some $x$, then $X \cong \mathbb{P}^n$ (Murayama, 2017).

7. Methods of Proof and Technical Innovations

Recent advances utilize order-of-vanishing methods, effective divisors with large multiplicity at a point, intersection-inequality technology, and bend-and-break to isolate exceptional curves. Particularly, the decomposition of adjoint divisors (e.g., $K_S+4A$ as a convex combination of other divisors and effective divisors with high vanishing) gives uniform bounds independent of classification results and characteristic (Rösler, 22 Jan 2026).

In Frobenius settings, the crucial tool is the finite flat Frobenius morphism and its trace map acting on global sections of canonical/adjoin bundles. Commutative diagrams involving global sections and appropriate jet-ideal sheaves yield powerful surjectivity results, establishing global generation and very ampleness via purely algebraic means (Mustata et al., 2012, Murayama, 2017).

These advances both strengthen lower bounds and unify geometric, cohomological, and arithmetic perspectives on the local positivity of adjoint divisors.