Maximal Multipoint Seshadri Constants in Higher Dimensions

1. A flag supplies the initial simplex of the correct volume

Choose $D$ with $DL$ very ample and a smooth integral complete-intersection flag cut out by $n-1$ general sections. Divide sections successively along the flag and bound the final vanishing order by the degree on its last curve. A lexicographic initial exponent $\alpha$ then satisfies

Thus $S_k\subset k\Delta(1/D,\ldots,1/D,D^{n-1}V)$ and $\#S_k=h^0(X,kL)=Vk^n/n!+O(k^{n-1})$, while multiplicativity of least terms gives $S_k+S_l\subset S_{k+l}$. At each fixed degree, choose positive weights exposing the finitely many least terms and shrink local coordinates. Persistence of a nonzero maximal jet minor transfers surjectivity from $M(S_k)$ to $H^0(X,kL)$. The weights may depend on the degree, but the flag, order and additive support family remain fixed.

Proposition 3.1 · pp. 8–9

2. One threshold works for every subsequent point count

Let $\ell>0$ be the smallest intercept of the initial simplex. Before fixing $r$, choose $r_0$ so that $w=(V/r)^{1/n}<\ell/100$ for every $r\geq r_0$. Proposition 4.1 replaces $(A,B)$ by $(w,AB/w)$; carry the large new intercept into the next pair. Every input pair has $A,B\geq\ell$, so $w\max(3/A,2/B)\leq3w/\ell<1/16$ and the proposition remains applicable. After $n-1$ replacements, the product of intercepts is still $V$, and

No restriction to square point counts or another subsequence occurs.

§6.1, equations (6.1)–(6.3) · pp. 16–17

3. Turn density into prescribed exponents and interpolate

Since $\operatorname{vol}(P_*)=V/n!$, only $o(k^n)$ lattice points are missing. Density alone does not guarantee a particular exponent. In Lemma 5.1, each $y\in kK\cap\mathbb Z^n$, for a fixed compact $K$ inside the simplex, has $\gg k^n$ decompositions $y=y_1+y_2$ into degrees $k_1=\lfloor k/2\rfloor$ and $k_2=k-k_1$. The holes exclude only $o(k^n)$ candidates. Some pair lies in $F_{k_1}\times F_{k_2}$, and additivity puts $y$ in $F_k$. No boundary saturation is asserted.

Lemma 5.1 · pp. 14–15

Given $0<\theta<w$, choose $\theta/w<\lambda<1$ and translate $\lambda P_*$ into the interior. For large $k$, its lattice points lie in $F_k$. The corresponding monomial translation acts by a unit on jet algebras at torus points. With $m=\lfloor k\theta\rfloor$, the set

lies in $k\lambda P_*$. Choose distinct $c_i\in\mathbb C^*$. For each coefficient of $(x'-1)^\alpha$, apply the Chinese remainder theorem modulo $(Y-c_i)^{m+1-|\alpha|}$. This prescribes arbitrary full jets through degree $m$ at $(1,\ldots,1,c_i)$. Proposition 5.3 then gives surjectivity, which transfers backward through every reshaping step to some distinct tuple on $X$. That tuple may depend on $\theta$ and $k$.

Lemma 5.2; Proposition 5.3 · pp. 15–16

1. A node gives a quadrilateral of equal area

Use $g=y^2-x^2(1+x)$ and local coordinates with $g=\xi\eta$. Write $f=g^jf_0$. On the normalization $x=v^2-1,y=v(v^2-1)$, the residual polynomial has degree at most $(\kappa-jd)W$, where $W=\max(2A,3B)=dH$. Its orders at the two branches $v=\pm1$ show that its least $(1,t)$-weight $\mu$ satisfies $(1+1/t)\mu\leq(\kappa-jd)W$. Restoring the least exponent $(j,j)$ of $g^j$ puts the least exponent of $f$ in

For $t>T=d^2H$, this quadrilateral has area $H/2$. Least-term bases in each label and compatibility of products preserve both cardinality and additivity across degrees.

§4, equations (4.3)–(4.13) · pp. 10–11

2. Tune a primitive direction to the exact required width

Apply the shear $(\alpha,\beta)\mapsto(\alpha-\beta,\beta)$ and compress by vanishing orders in the second coordinate. The result lies in $P=\operatorname{conv}\{(-b,0),(a,0),(0,c)\}$. Put $R=H/w$ and choose $j$ as the largest power of two at most $R/(2W)$. The inequality $dw<1/16$ makes the interval $(d(R-jW),d(R-ja_0))$, with $a_0=WT/(1+T)$, longer than $2$. Choose an odd integer $i$ in it and then $t>T$ with $i/d+ja=R$. The vector $u=(i,j)$ is primitive, and the range of $\det(u,\cdot)$ on $P$ is exactly $R$.

Extend $u$ to an integral basis. Horizontal slice lengths form a tent, whose area $H/2$ and height range $R$ force its maximum width to be $H/R=w$. Horizontal and then vertical compression yield $\Delta(w,R)$. No rationality of the intercepts is required. Compression does not merely translate lattice points: it replaces each row of Laurent polynomials by its possible vanishing orders at $z=1$.

§4, equations (4.14)–(4.19) · pp. 12–14

3. Separate fixed supports from degree-dependent jet transfers

The initial order, integral changes and exact vanishing orders defining compression are fixed in all degrees and preserve multiplication of labelled spaces. Small analytic scaling parameters and weights exposing tie-breaks may instead depend on the finite degree and test tuple. Convergence of derivatives and persistence of a nonzero finite minor transfer surjectivity backward for full jets, including the unchanged $n-2$ variables. This distinction preserves both the graded structure and the interpolation implications under iteration.

Lemmas 2.2–2.7 · pp. 5–7

1. From special test tuples to a common very-general locus

For fixed integers $k,m$, surjectivity onto jets is Zariski open: smoothness makes principal parts locally free, and maximal minors define the surjectivity locus. Choose $\theta_j\uparrow w$. For each $j$, all $k\geq k_j$ yield nonempty open sets $O_{j,k}\subset U_r(X)$. Remove $Z_r=\bigcup_j\bigcup_{k\geq k_j}(U_r(X)\setminus O_{j,k})$. The complex space $U_r(X)(\mathbb C)$ is locally compact Hausdorff and hence Baire; each proper algebraic closed subset has empty complex interior. Thus the complement is nonempty. The constructed special torus tuples are not themselves called general: nonzero rank conditions and a countable intersection provide that passage.

§6.2, equation (6.4) · p. 17

2. Full jets give the lower bound with ordinary multiplicities

Lemma 6.1 deduces $A\cdot C\geq m\sum_i\operatorname{mult}_{p_i}C$ from separation of full jets through degree $m\geq1$ by $A$. Lift a nonzero element of $\mathfrak m_C^m/\mathfrak m_C^{m+1}$ at one marked point and prescribe zero jets at the others. The resulting section does not vanish identically on $C$. Even when $C$ is singular, for nonzero $f\in\mathfrak m_C^m$,

bounds the local intersection by $m$ times the ordinary Hilbert–Samuel multiplicity. Take $A=kL$ and $m=\lfloor k\theta_j\rfloor$. First letting $k\to\infty$ and then $j\to\infty$ gives $\varepsilon\geq w$.

Lemma 6.1; equation (6.6) · p. 18

3. The blow-up upper bound and the nef-class conclusion

For any distinct tuple, put $D_t=\pi^*L-t\sum E_i$. Curves inside $E_i$ have nonnegative intersection because $D_t|_{E_i}=t\mathcal O_{\mathbb P^{n-1}}(1)$. Every other curve $\Gamma$ is the strict transform of an integral curve $C\subset X$, and

The manuscript explains $E_i\cdot\Gamma=\operatorname{mult}_{p_i}C$, including for singular $C$, using the blow-up of its maximal ideal and the Hilbert–Samuel multiplicity of the exceptional fibre. Hence $D_t$ is nef for $0\leq t<\varepsilon$, and

gives the universal upper bound $\varepsilon\leq w$. Combine it with Step 2 for equality. Substituting the lower bound into every curve intersection proves that $D_w$ is nef, and the intersection computation gives $D_w^n=0$. Conversely, nefness of $D_w$ gives $\varepsilon\geq w$ on strict transforms; the upper bound then gives equality. Zero top self-intersection alone is not used to infer nefness.

§6.4, equations (6.7)–(6.8) · p. 19