Maximal multipoint Seshadri constants in positive characteristic

1. Reshape intercepts while keeping one graded structure

Choose $D$ with $DL$ very ample and a smooth complete-intersection flag. At each Bertini step the variety being cut has dimension at least two, and the morphism induced by $DL$ is an embedding. Lexicographically least exponents in adapted coordinates give

Here $\Delta(a_1,\ldots,a_n)=\{x_i\geq0:\sum_i x_i/a_i\leq1\}$. The containment follows by successively dividing by flag equations and then bounding the final vanishing order by the degree of a zero divisor on the terminal curve. Apply Proposition 4.1 $n-1$ times, preserving the product of intercepts and the cardinality in each degree, to reach $\Delta(w,\ldots,w,rw)$. At every stage, jet separation after transformation transfers back to the preceding space.

Proposition 3.1 · pp. 7–8

Proposition 4.1 · pp. 8–11

2. A lower bound from jets and a separate volume upper bound

For each fixed $0<\theta<w$, Proposition 5.3 supplies jets through order $\lfloor N\theta\rfloor$ at the geometric generic tuple for every sufficiently large $N$. In Lemma 6.1 one prescribes a nonzero class in $\mathfrak m^m/\mathfrak m^{m+1}$ of one curve local ring, and zero jets at the other points. The resulting section vanishes to order at least $m$ at every marked point while remaining nonzero on the curve. The local inequality $\operatorname{length}(R/(f))\geq m e(R)$ then gives, including for singular integral curves,

Let $N\to\infty$ and then $\theta\uparrow w$ to obtain $\varepsilon_r\geq w$.

Separately, for rational $0\leq s<\varepsilon_r$, consider $D_s=\pi^*L_{K_r}-s\sum E_i$ on the blow-up. Curves inside $E_i$ are handled by $D_s|_{E_i}=s\mathcal O_{\mathbf P^{n-1}}(1)$; other curves by $E_i\cdot\widetilde C=\operatorname{mult}_{p_i}C$. Thus $D_s$ is nef and $0\leq D_s^n=V-rs^n$, giving $\varepsilon_r\leq w$. The upper bound alone does not establish maximality: it must meet the lower bound constructed from jets.

Proposition 5.3 · pp. 13–14

Lemma 6.1 · p. 15

Lemma 6.2 · pp. 15–16

3. The boundary class and dimension two

Closedness of the nef cone and equality of the bounds imply that $D_w$ is nef with $D_w^n=0$. This connection uses the curve definition and the blow-up intersection calculation, rather than only a numerical correspondence. Remark 6.3 observes that the flag, reshaping, interpolation and intersection arguments all apply when $n=2$, with only one reshaping step. Strict positivity on every curve is not asserted here.

Proof of Theorem 1.1; Remark 6.3 · p. 16

1. Bound exponents using the two branches

Use $g=y^2+xy-x^3$. Hensel lifting the simple roots $0,-1$ of $Z^2+Z-x$ gives formal coordinates with $g=\xi\eta$. The roots remain distinct when $p=2$, since $-1=1\ne0$. Under $x=v(v+1),\ y=v^2(v+1)$, the sum of the orders at $v=0,-1$ of a factor not divisible by $g$ is bounded by its degree. Consequently its least exponent for the order $(\alpha+t\beta,\beta)$ lies in a quadrilateral $Q$ of area $H/2$.

Formal coordinates alone do not transfer interpolation. Lemma 2.3 chooses common integral weights realizing the finitely many initial terms in each degree. Lemma 2.4 substitutes $z_a=s^{\lambda_a}(Q_a+\delta_a)$ into truncated jet rings. After column normalization, the evaluation matrix specializes at $s=0$ to the monomial matrix. A nonzero maximal minor remains nonzero over $k((s))$. Setting $\delta=0$ defines actual $k((s))$-points, and the substitution factors through their order-$m$ jet algebras. Surjectivity of the matrix and equality of dimensions make these algebra maps isomorphisms. Repeated centers would contradict surjectivity, so separation occurs at distinct points and the corresponding open locus is nonempty. The auxiliary weights may depend on the degree, but the formal coordinates and monomial order defining the graded family stay fixed.

Lemma 4.2 · p. 9

Lemmas 2.2–2.4 · pp. 4–6

2. Fix the primitive direction and compressions in every degree

A shear and compression turn $Q$ into a triangle of the same area. Lemma 4.3 chooses $t$ and a primitive vector $u=(i,j)$ so that the height range under $z\mapsto\det(u,z)$ is exactly $H/w$. Choosing $j$ as a power of two and $i$ as an odd integer in a suitable open interval provides both primitivity and the required real intercept. After an integral basis change, horizontal and vertical compressions give the bounding triangle $\Delta(w,H/w)$. The other coordinates remain labels.

Compression does not fill every integer in a slice. In characteristic $p$, the orders of $\langle1,z^p\rangle$ at $z=1$ are $0,p$, leaving gaps. Lemma 2.6 preserves the interval bound, cardinality, gradedness and backward transfer. Filling every interior lattice point requires the separate argument in the next section.

Lemma 4.3 · pp. 10–11

Lemmas 2.5–2.6 · p. 6

1. Turn density into the presence of individual lattice points

Let $\ell$ be the smallest initial intercept and choose $r_0$ so that $w<\ell/100$. At each reshaping step the processed intercepts $A,B$ are at least $\ell$, hence $w\max(3/A,2/B)<1/16$. The final family $F_N$ lies in $P_* =\Delta(w,\ldots,w,rw)$ and has $\#F_N=\operatorname{vol}(P_*)N^n+o(N^n)$.

Lemma 5.1 combines this density with additivity. Each lattice point $y$ in a fixed compact interior region has at least $cN^n$ decompositions $y=y_1+y_2$ in nearly equal degrees. Missing lattice points exclude only $o(N^n)$ decompositions. One remains with both summands in the family, so $y\in F_N$. Density alone is not being identified with pointwise filling.

Lemma 5.1 · p. 12

2. Interpolate in the elongated simplex and transfer to a nonempty open set

For $m=\lfloor N\theta\rfloor$, set

Choose distinct nonzero $c_i\in k$ and express arbitrary jets at $q_i=(1,\ldots,1,c_i)$ as a sum of terms $(x'-1)^\alpha h_\alpha(Y)$. The Chinese remainder theorem supplies $h_\alpha$ with $\deg h_\alpha<r(m-|\alpha|+1)$, so all monomials lie in $E_m$. Neither factorials nor ordinary derivatives are required.

Choose $\theta/w<\lambda<1$ and a small positive vector $v$ such that $v+\lambda P_*\subset\operatorname{int}P_*$. An integral translate puts $E_m$ inside the filled region; multiplication by a Laurent monomial is a unit on the torus and preserves jet separation. Backward transfer returns to some tuple on $X$. For fixed $r,\theta,N$, surjectivity of evaluation into the principal-parts bundles is an open rank condition on $U_r$.

Lemma 5.2 · p. 13

Proposition 5.3 · pp. 13–14

3. Impose every condition at the same geometric generic tuple

The generic point of integral $U_r$ belongs to each nonempty open locus, and base change for principal parts respects jet evaluation. Thus the same $K_r$-tuple satisfies all required separations. No $k$-point needs to be chosen in a countable intersection of open sets, so countable algebraically closed fields are included. This conclusion must not be replaced by a statement for arbitrary general closed points over $k$.

Proposition 5.3 · pp. 13–14