--- title: Newton Polygons of Curves url: https://www.emergentmind.com/topics/newton-polygons-of-curves type: topic --- # Newton Polygons of Curves Newton polygons of curves arise in several adjacent but distinct senses. For smooth projective curves over a field of characteristic \(p\), the dominant arithmetic meaning is the Newton polygon of the Jacobian, equivalently the slope polygon of Frobenius on \(H^1_{\mathrm{cris}}\) or of the \(p\)-divisible group \(J(C)[p^\infty]\); its realizability inside the Torelli locus is a central question in the geometry of \(\mathcal A_g\). In toric and singularity theory, by contrast, the Newton polygon is the convex hull of exponent vectors of a Laurent polynomial or plane curve germ, and its faces govern genus, gonality, topology, multiplier ideals, and jet schemes. The modern literature therefore treats “Newton polygons of curves” as a family of related constructions linking arithmetic geometry, toric geometry, singularity theory, and \(p\)-adic analysis [2509.00998] [1106.3762] [2509.06095]. ## 1. Basic notions and competing meanings The phrase “Newton polygon of a curve” is used in at least four standard ways. | Meaning | Input | Output | |---|---|---| | Jacobian Newton polygon | smooth projective \(C/k\), \(\operatorname{char} k=p\) | slopes of Frobenius on \(J(C)\) | | Newton polygon of an equation | Laurent polynomial \(f(x,y)\) | lattice polygon \(\Delta(f)\subset \mathbb R^2\) | | Newton polygon of a singularity | plane curve germ \(f(x,y)\) at the origin | compact faces of \(\Gamma(f,x,y)\) and dual fan | | Convergence Newton polygon | differential equation on a Berkovich curve | slopes \(-\log R_{S,i}(x)\) of the multiradius | For a principally polarized abelian variety \(A\) of dimension \(g\) in characteristic \(p\), the Newton polygon has height \(2g\), endpoints \((0,0)\) and \((2g,g)\), and slopes \(m_1,\dots,m_{2g}\in [0,1]\cap \mathbb Q\) satisfying the symmetry \(m_i+m_{2g+1-i}=1\). For a curve \(C\), one defines \(\operatorname{NP}(C):=\operatorname{NP}(J(C))\). The \(p\)-rank equals the multiplicity of slope \(0\), and ordinary and supersingular Jacobians correspond respectively to slope multisets \(\{0^g,1^g\}\) and \(\{(1/2)^{2g}\}\) [2006.04927]. For a bivariate Laurent polynomial \[ f(x,y)=\sum_{(i,j)\in\mathbb Z^2} a_{ij}x^iy^j, \] the Newton polygon is \[ \Delta(f):=\operatorname{conv}\{(i,j)\in\mathbb Z^2\mid a_{ij}\neq 0\}\subset \mathbb R^2. \] Under the standard nondegeneracy condition, the toric closure of \(U(f)=\{f=0\}\subset (\mathbb C^*)^2\) is smooth, and the interior polygon \(\Delta^{(1)}\) controls the canonical model and genus. For a plane curve singularity \(f(0,0)=0\), the Newton polygon \(\Gamma(f,x,y)\) is the union of the compact faces of the Newton polyhedron at the origin, and its dual Newton fan organizes face polynomials and Newton non-degeneracy. In the Berkovich setting, the convergence Newton polygon at \(x\) has slopes \(s_i(x)=-\log R_{S,i}(x)\), where \(R_{S,i}\) are the normalized radii of convergence of horizontal sections [1106.3762] [2509.06095] [1209.3663]. A persistent source of ambiguity is that these notions are not interchangeable. The Newton polygon of \(J(C)\) is an isogeny invariant in characteristic \(p\), whereas \(\Delta(f)\) and \(\Gamma(f,x,y)\) are combinatorial invariants of defining equations and coordinate systems. Much of the current literature is devoted precisely to translating between them when a curve is given by explicit covers or toric models. ## 2. Jacobians, slopes, and moduli For a smooth projective geometrically irreducible curve \(C/k\) of genus \(g\) over an algebraically closed field of characteristic \(p>0\), the Jacobian Newton polygon can be defined either from crystalline cohomology \(H^1_{\mathrm{cris}}(J(C)/W(k))\otimes K\) with Frobenius \(\varphi\), or from the Dieudonné–Manin decomposition of \(J(C)[p^\infty]\). The slopes \(\lambda_1,\dots,\lambda_{2g}\) satisfy \[ \lambda_i\in[0,1],\qquad \lambda_i+\lambda_{2g+1-i}=1,\qquad \sum_{i=1}^{2g}\lambda_i=g. \] The \(p\)-rank \(f\) is the multiplicity of slope \(0\), and the \(a\)-number is \(a=\dim_k\operatorname{Hom}(\alpha_p,J(C)[p])\). If \(\beta=\{\omega_1,\dots,\omega_g\}\) is a basis of \(H^0(C,\Omega^1_C)\) and \(M\) is the Cartier–Manin matrix, then \[ a=g-\operatorname{rank}(M), \] while \(f\) is the stable rank of \[ M^{(p^{g-1})}M^{(p^{g-2})}\cdots M^{(p)}M. \] All symmetric Newton polygons occur for principally polarized abelian varieties, but the Torelli problem asks which of them occur for Jacobians [2509.00998]. This moduli-theoretic formulation is expressed by the Torelli morphism \(\tau_g:\mathcal M_g\to\mathcal A_g\). The open Torelli locus \(T_g^\circ=\tau_g(\mathcal M_g)\) sits inside \(\mathcal A_g\), and for \(g\ge 4\) one has \(\dim T_g=3g-3<\dim \mathcal A_g=g(g+1)/2\). Newton polygon strata \(\mathcal A_g[\xi]\) and \(p\)-rank strata \(\mathcal M_g^f\) therefore intersect the Torelli locus in a highly constrained way. Every irreducible component of \(\mathcal M_g^f\) has dimension \(2g-3+f\), so the \(p\)-rank stratification is completely nonempty, but the finer Newton polygon stratification remains only partially understood. A standard misconception is that Ekedahl–Oort type refines Newton polygon uniformly; in genus \(3\), EO type \(I_{3,1}\) can have Newton polygon with slopes \(\{1/3,2/3\}\) or can be supersingular, so neither stratification dominates the other [1806.04654]. Known realizability results in small genus are already nontrivial. In genus \(2\), the three Newton polygons of abelian surfaces occur for Jacobians of genus-\(2\) curves for all \(p\), except the superspecial case when \(p=2,3\). In genus \(3\), all five Newton polygons of abelian threefolds occur for Jacobians over \(\overline{\mathbb F}_p\), with some EO-type exceptions in characteristic \(2\). In genus \(4\), supersingular curves exist for every prime \(p\), and several further polygons with slopes \(1/4,3/4\) or \(\{1/3,1/2,2/3\}\) are known to occur [2509.00998]. ## 3. Torelli loci, Newton strata, and unlikely intersections The modern arithmetic formulation asks which Newton strata intersect the Torelli locus. For a fixed Newton polygon \(P\) of height \(2g\), let \(W_P\subset \mathcal A_g\) be the closed locus where \(\operatorname{NP}_x\succeq P\), and \(W_P^0\subset W_P\) the open locus where \(\operatorname{NP}_x=P\). The intersection problem is whether \(W_P^0\cap T_g\) or \(W_P\cap T_g\) is nonempty. Oort’s lattice-count theorem identifies the codimension of \(W_P\) in \(\mathcal A_g\) with the number of lattice points strictly below \(P\) when the vertices are integral, and this turns many Torelli questions into explicit codimension estimates [2006.04927]. A central existence theorem of Kramer–Miller gives large families of Jacobians with many slope-\(1/2\) segments. If \(p\) is an odd prime and \(k\) satisfies \[ \frac{2g}{3}-\frac{2p(p-1)}{3}\ge 2k(p-1), \] then there exists a curve \(C_{g,k}\) of genus \(g\) whose Newton polygon has slope multiset \[ \{0,1\}^{g-k(p-1)}\sqcup \{1/2\}^{2k(p-1)}. \] More generally, if \(d\ge 2\) with \(p\nmid d\), \(\delta=1\) for \(d\) odd and \(\delta=2\) for \(d\) even, and \[ \frac{2g}{d+1}-\frac{2p(p-1)}{d+1}\ge k\delta(p-1), \] then there exists \(C_{g,k}\) with \[ \operatorname{NP}(C_{g,k})\succeq \{0,1\}^{g-\frac{k\delta(p-1)(d-1)}{2}} \sqcup \{1/d,2/d,\dots,(d-1)/d\}^{k\delta(p-1)}, \] and if \(p\equiv 1 \pmod d\) this can be taken as an equality. These curves arise from \(\mathbb Z/p\mathbb Z\)-covers with controlled Swan conductors and exhibit intersections of \(T_g\) with Newton strata whose codimensions are too large to be predicted by naive dimension counts, hence “unlikely intersections” [2006.04927]. The same paper constructs full families \(\{C_g\}_{g\ge 1}\) with asymptotic lower bounds on scaled Newton polygons. Writing \(s\operatorname{NP}(C):=(1/g)\operatorname{NP}(C)\), there exists a family such that \[ s\operatorname{NP}(C_g)\succeq P(x^2/4,2) \] for all large \(g\), equivalently \[ \operatorname{NP}(C_g)\succeq y=\frac{x^2}{4g} \] in unscaled coordinates. This forces many slopes away from \(0\) and \(1\) and yields an unlikely family in the Torelli locus. The same framework supplies evidence for Oort’s amalgamation conjecture: if two curves realize Newton polygons \(P\) and \(P'\), then one expects a curve of genus \(g+g'\) with Newton polygon \(P\sqcup P'\); within the Kramer–Miller families this closure under disjoint union is proved in explicit congruence ranges [2006.04927]. ## 4. Covers, explicit constructions, and Oort’s conjecture Artin–Schreier and cyclic covers provide the main explicit source of Jacobians with prescribed Newton polygons. For an Artin–Schreier cover \(\pi:Y\to X\) of degree \(p\) with branch locus \(B\) and lower jumps \(D=\{d_Q\}_{Q\in B}\), Booher–Pries define two lower bounds, \(\operatorname{NP}_{\mathrm{Hodge}(D)}\) and the sharper \(\operatorname{NP}_X(D)\), assembled from \(\operatorname{NP}(X)\), ordinary slope blocks, and local generic polygons \(\mathrm{GNP}(d_Q,p)\). If \(X\) is ordinary and \(p>\max\{3d_Q\}\), then there exists such a cover with \[ \operatorname{NP}_{\mathrm{Hodge}(D)}\le \operatorname{NP}(Y)\le \operatorname{NP}_X(D). \] If moreover \(p\equiv 1\pmod{d_Q}\) for every \(Q\in B\), then \(\operatorname{NP}_{\mathrm{Hodge}(D)}=\operatorname{NP}_X(D)\), so one obtains a cover with minimal Newton polygon in the partial order. The proof combines formal patching with specialization from a singular cover whose Jacobian Newton polygon can be computed exactly [2003.09027]. For cyclic covers of \(\mathbf P^1\) branched at three points, the Shimura–Taniyama method turns Frobenius orbits on \(\operatorname{Hom}(\mathbb Q[\mu_m],\mathbb C)\) into explicit slope formulas. This produces numerous concrete Jacobian Newton polygons. Under congruence conditions on \(p\), the paper on cyclic three-point covers realizes the supersingular polygon for each genus \(g\) with \(4\le g\le 11\), nine nonsupersingular polygons of \(p\)-rank \(0\) for \(4\le g\le 11\), and for all \(g\ge 5\) the polygon with \(p\)-rank \(g-5\) having slopes \(1/5\) and \(4/5\) [1805.04598]. A broad geometric existence theorem was later obtained by Pries. For every prime \(p\) and every \(g\ge 4\), every Newton polygon whose \(p\)-rank is at least \(g-4\) occurs for a smooth curve of genus \(g\). The same paper gives a new proof that supersingular curves of genus \(4\) exist for every prime \(p\), proves that every symmetric Newton polygon in dimension \(4\) occurs on \(\mathcal M_4\), and resolves cases of Oort’s conjecture by showing that if \(\nu_d^0\) occurs in genus \(d\), then \(\nu_d^0\oplus \mathrm{ss}\) occurs in genus \(d+1\) [2306.11080]. The most recent explicit constructions exploit abelian covers of \(\mathbf P^1\) branched at three points. For genera \(9\le g\le 50\), computations in these families yield natural densities \[ \delta_{ss}(g)>0.7,\qquad \delta_{ssp}(g)>0.2,\qquad \delta_{nu}(g)>0.875 \] for the existence of supersingular curves, superspecial curves, and curves with unlikely Newton polygons. The same framework gives a genus-\(12\) curve with only slopes \(5/12\) and \(7/12\) in odd characteristics \[ p\equiv 3,12,17,33 \pmod{35}, \] produces new supersingular curves of arbitrarily large genus over fixed odd primes, and supplies further evidence for Oort’s conjecture in cyclic three-point families [2602.07693]. ## 5. Combinatorial, toric, and local Newton polygons When a curve is defined by a Laurent polynomial, the Newton polygon controls classical birational invariants. For a nondegenerate \(f\in \mathbb C[x^{\pm1},y^{\pm1}]\), the geometric genus of \(U(f)\) is \[ g(U(f))=\big|\Delta^\circ\cap \mathbb Z^2\big|, \] and the canonical model is governed by the interior polygon \(\Delta^{(1)}\). The gonality satisfies \[ \operatorname{gon}(U(f))\le \operatorname{lw}(\Delta(f)), \] with the refined bound \[ \operatorname{gon}(U(f))\le \operatorname{lw}(\Delta(f))-1 \] for the exceptional shapes \(\Delta\cong d\Sigma\) or \(\Delta\cong 2\Upsilon\). A key identity is \[ \operatorname{lw}(\Delta)=\operatorname{lw}(\Delta^{(1)})+2, \] unless \(\Delta\cong d\Sigma\), in which case \(\operatorname{lw}(\Delta)=\operatorname{lw}(\Delta^{(1)})+3=d\). The generic sharpness conjecture asserts that these combinatorial bounds are generically attained [1106.3762]. For smooth curves on \(\mathbf P^1\times \mathbf P^1\), the interior polygon can become intrinsic. If \(C\subset \mathbf P^1\times \mathbf P^1\) is a smooth projective curve of genus \(g\neq 4\) and is birational to a \(\Delta\)-nondegenerate curve, then \(\Delta^{(1)}\) is, up to unimodular transformation, a standard rectangle \[ [0,\alpha]\times[0,\beta]. \] The same paper shows that first scrollar Betti numbers can be read directly from the row structure of \(\Delta^{(1)}\), under mild combinatorial conditions [1304.4997]. The Newton polygon also governs enumerative geometry of generic complex plane curves with a fixed support polygon \(P\). If \(C\subset (\mathbb C^*)^2\) is generic with Newton polygon \(P\), then the number of inflection points is \[ \#(\mathrm{infl})= 6\operatorname{vol}(P) -2\sum_{\gamma\in\{\gamma_\downarrow,\gamma_\Delta,\gamma_\leftarrow\}}\operatorname{len}(P^\gamma) -\sum_{\gamma\in\{\gamma_\uparrow,\gamma_\nabla,\gamma_\rightarrow\}}\operatorname{len}(P^\gamma), \] and the number of bitangents is \[ \#(\mathrm{bitang})= -10\operatorname{vol}(P)+\operatorname{vol}(P^\vee) +3\sum_{\gamma\in\{\gamma_\downarrow,\gamma_\Delta,\gamma_\leftarrow\}}\operatorname{len}(P^\gamma) +\sum_{\gamma\in\{\gamma_\uparrow,\gamma_\nabla,\gamma_\rightarrow\}}\operatorname{len}(P^\gamma). \] If \(P\supset 5\Delta\), then the projectively dual curve \(C^\vee\) has no singularities other than nodes and cusps [2204.04626]. At the singularity level, Newton polygons control topology, jet schemes, and multiplier ideals. For Newton non-degenerate plane curve singularities, the paper on jet schemes constructs a staircase subgraph \(J_{SC}(f)\) from lattice walks determined by continued fractions \(SC(q/p)\), proves that the irreducible components of jet schemes are encoded by this graph, and shows that the full graph of jet components determines the embedded topological type [2509.06095]. For multiplier ideals, a toroidal embedded resolution built by iterated regularized Newton modifications yields a finite collection of local Newton polygons \(\mathcal N_{R_i,L_i}(C)\), and the multiplier ideal is characterized by \[ \mathcal J(\xi C)_o= \big\{ h\in \mathcal O_{S,o}\mid \mathcal N_{R_i,L_i}(C_h)+\underline\lambda_i\subset \operatorname{Int}\big(\xi\,\mathcal N_{R_i,L_i}(C)\big) \ \forall i \big\}. \] This generalizes Howald’s formula from monomial ideals and Newton non-degenerate settings to arbitrary plane curve singularities [2109.13294]. ## 6. Variation in families and analytic avatars Newton polygons also govern variation phenomena. Over a finite field, every convergent \(F\)-isocrystal on an abelian variety has constant Newton polygon. Applied to a proper smooth family of connected curves \(f:X\to S\), this implies that if \(S\) is an abelian variety, then the relative \(F\)-isocrystal \(R^1f_{\mathrm{rig},*}\mathcal O_X\) has constant Newton polygons, and the family is isotrivial. More generally, if every geometric convergent \(F\)-isocrystal on a projective smooth base \(S\) has constant Newton polygons, then every proper smooth family of genus-\(\ge 2\) curves over \(S\) is isotrivial [1704.00856]. For abelian \(L\)-functions on curves, the Newton polygon can be assembled from local data. If \(X\) is a smooth affine curve over \(\mathbb F_q\) and \(\rho:\pi_1(X)\to \mathbb C_p^\times\) is a finite character of order \(p^n\), then the global Hodge polygon has slope multiset \[ \{0^{\,g-1+|S|}\}\sqcup \{1^{\,g-1+|S|}\}\sqcup \bigsqcup_{P\in S}\{1/d_P,\dots,(d_P-1)/d_P\}, \] where \(d_P\) are the Swan conductors. When \(\overline X\) is ordinary, the truncated polygons \(\operatorname{NP}_q^{