Singularities of rational maps: foundations and surfaces

Abstract: We develop a theory of singularities of rational maps, focusing on maps $f\colon X\dashrightarrow \mathbb P^n,$ and using their polarised graphs and normalised polarised graphs even when the source $X$ is very singular. To measure singularities of the map at a point $x\in X$, i.e. how far it is from being regular, we introduce invariants including the normalised graph fibre degree $δ_x(f)$, a generalised lc threshold $λ_x(f)$, and invariants measuring the singularities of the normalised graph itself. Numerous examples show that the resulting invariants measure genuinely different aspects of map singularities. We investigate the surface case in detail. We relate the normalised graph fibre degree to multiplicity, prove a sharp threshold--degree inequality for klt surface germs, and develop a detailed theory of linear type maps on smooth and singular surfaces. In particular, we connect the existence of linear type maps to existence of smooth curves through the given point, and with local class groups and complement theory. We conclude with questions and future directions concerning higher dimensions, complements and boundedness, moduli, Cremona groups, commutative algebra, curve-counting theories, and positive characteristic.

• The paper introduces three independent invariants—normalised graph fibre degree, generalised lc threshold, and normalised graph mld—to distinguish the geometry of a rational map from singularities of its source.
• The paper proves the sharp inequality δₓ(f)λₓ(f) ≤ 2 for singular maps from klt surface germs, identifies fibre degree with classical multiplicity, and establishes boundedness for normalised graphs of maps from smooth surfaces.
• The paper characterises linear-type maps on smooth and several singular surfaces, linking them to canonical graph singularities, smooth curves, local class groups, toric geometry, and reduced 2-complements while highlighting unresolved higher-dimensional questions.

Motivation and framework

The paper develops the foundations of a theory measuring how singular a rational map $f\colon X\dashrightarrow \mathbb P^n$ is at a point $x\in X$, i.e. how far $f$ is from being a morphism at $x$. The author's central methodological commitment is that the singularities of the source variety and those of the map are independent data: a map from a smooth variety may be highly singular, while a map from a deeply singular variety may be mild. Accordingly, the basic constructions are formulated for normal varieties or arbitrary normal surface germs, with klt, rational, or smooth hypotheses imposed only when the birational tools demand them.

The main geometric object is the polarised graph $Y=\Gamma_f\subset X\times\mathbb P^n$ (with reduced structure), together with the tautological polarisation $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$, and, since the graph is frequently non-normal even over smooth sources, its normalisation $Y^\nu$ with projections $p\colon Y^\nu\to X$, $q\colon Y^\nu\to\mathbb P^n$. The polarisation is essential: the birational morphism $Y\to X$ alone does not determine the singularity of the map. When $x\in X$0 is given locally by a base ideal $x\in X$1, the graph is the blowup of $x\in X$2, governed by the Rees algebra; the polarised graph formulation has the advantage of remaining intrinsic at singular points where no preferred base ideal exists.

Invariants of map singularities

Three families of invariants are introduced. The normalised graph fibre degree $x\in X$3 measures the polarised fibre; refined data is captured by the Hilbert polynomial of $x\in X$4 and by the degree spectrum $x\in X$5 of degrees of cycles supported on the fibre. The generalised lc threshold $x\in X$6 is defined by viewing $x\in X$7 as a generalised pair whose nef part is induced by $x\in X$8, for a hyperplane $x\in X$9; it is independent of $f$0, agrees at smooth points with the log canonical threshold of the primitive local base ideal, and satisfies the ACC by the known ACC for generalised lc thresholds. Finally, the normalised graph mld $f$1 measures the intrinsic singularities created on $f$2 itself. The examples demonstrate that these invariants are genuinely independent: maps on $f$3 are constructed with $f$4 and $f$5 fixed while $f$6 or $f$7 tends to zero, and a toric degree-one example has both $f$8 and $f$9 of arbitrarily deep klt type while $x$0 is linear type.

Comparison with multiplicity and the sharp threshold–degree inequality

The first structural result identifies the fibre degree with classical multiplicity: for a normal surface and a general hyperplane $x$1, $x$2, proved by reducing to the fact that for reduced curves the Hilbert–Samuel multiplicity equals the degree of the normalisation fibre. A toric formula gives $x$3 in terms of lattice data of the kernel ray of $x$4.

The central inequality states that for a klt surface germ and $x$5 singular at $x$6,

$x$7

and the constant $x$8 is sharp, attained by $x$9 with $Y=\Gamma_f\subset X\times\mathbb P^n$0 and $Y=\Gamma_f\subset X\times\mathbb P^n$1. The proof uses the anti-nefness of the maximal-ideal cycle $Y=\Gamma_f\subset X\times\mathbb P^n$2 on a resolution and the arithmetic genus computation $Y=\Gamma_f\subset X\times\mathbb P^n$3 via Lipman's rationality results. The inequality implies that a positive lower bound on the threshold bounds the fibre degree; the converse fails, as shown by an explicit family on $Y=\Gamma_f\subset X\times\mathbb P^n$4 singularities with $Y=\Gamma_f\subset X\times\mathbb P^n$5 but $Y=\Gamma_f\subset X\times\mathbb P^n$6. On smooth surfaces a stronger equality holds: $Y=\Gamma_f\subset X\times\mathbb P^n$7 for the generalised $Y=\Gamma_f\subset X\times\mathbb P^n$8-lc threshold, and the degree spectrum contains the inverse thresholds arising after successive point blowups.

Singularities of the normalised graph and boundedness

The fibre degree and threshold do not control the singularities of $Y=\Gamma_f\subset X\times\mathbb P^n$9: examples on $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$0 exhibit non-lc normalised graphs (a degree-5 map with coefficient $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$1 on the central $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$2-curve of the resolution). Conversely, a boundedness result holds: if $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$3 from a smooth surface is singular only at $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$4 with $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$5, then $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$6 is $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$7-bounded, meaning all exceptional curves on the minimal resolution have self-intersection at least $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$8 and at most $\mathcal O_Y(1)=g^*\mathcal O_{\mathbb P^n}(1)$9 have self-intersection $Y^\nu$0. The proof runs an MMP with scaling of $Y^\nu$1 and counts single versus double blowups.

Linear type maps

A map is of linear type at $Y^\nu$2 if $Y^\nu$3. On smooth surfaces, linear type is strikingly rigid: assuming $Y^\nu$4 singular at $Y^\nu$5, it is equivalent to each of the graph being normal with canonical singularities and reduced fibre $Y^\nu$6 of degree one; $Y^\nu$7 ample over $Y^\nu$8; $Y^\nu$9; and $p\colon Y^\nu\to X$0 smooth at $p\colon Y^\nu\to X$1 for general $p\colon Y^\nu\to X$2. In this case $p\colon Y^\nu\to X$3 has at most one singular point over $p\colon Y^\nu\to X$4, of type $p\colon Y^\nu\to X$5.

On singular surfaces the picture changes substantially. For smooth or rational singularities, linear type is equivalent to $p\colon Y^\nu\to X$6. For an arbitrary normal surface germ, existence of a map with $p\colon Y^\nu\to X$7 is equivalent to existence of a curve through $p\colon Y^\nu\to X$8 smooth at $p\colon Y^\nu\to X$9 (and one may take target $q\colon Y^\nu\to\mathbb P^n$0). Every affine toric surface germ admits a toric singular linear type map. Over $q\colon Y^\nu\to\mathbb P^n$1, a singular canonical surface germ admits a singular linear type map if and only if $q\colon Y^\nu\to\mathbb P^n$2; since algebraic realisations with a fixed completion can have arbitrary subgroup of the completed class group, existence of a linear type map is not determined by the completed local singularity. In particular, $q\colon Y^\nu\to\mathbb P^n$3 canonical germs admit none, which also follows directly from the fact that the maximal-ideal cycle has all coefficients at least $q\colon Y^\nu\to\mathbb P^n$4. The paper further develops the theory for $q\colon Y^\nu\to\mathbb P^n$5-type and $q\colon Y^\nu\to\mathbb P^n$6-type klt singularities, exhibiting a $q\colon Y^\nu\to\mathbb P^n$7-type klt germ with class group $q\colon Y^\nu\to\mathbb P^n$8 that admits no reduced $q\colon Y^\nu\to\mathbb P^n$9-complement, and raising the question whether existence of a singular linear type map is equivalent to existence of a reduced $Y\to X$0-complement for non-canonical $Y\to X$1-type klt singularities.

Limitations and open questions

The detailed theory is confined to dimension two; in higher dimensions even the fibre $Y\to X$2 need not be a projective space when $Y\to X$3 is smooth and $Y\to X$4 is smooth, and the case $Y\to X$5 is not classified even on surfaces. The termination of the threshold-decomposition procedure for birational maps via increasing generalised $Y\to X$6-lc thresholds relies on an ACC that is conjectural for $Y\to X$7 beyond dimension two. Several specific questions remain open: classification of germs admitting singular linear type maps in each dimension; boundedness of the pluricanonical map $Y\to X$8 on a klt Fano contraction in terms of map singularities; adjunction and inversion of adjunction relating singularities of $Y\to X$9 and of its restriction to $x\in X$00; behaviour of map singularities under composition; and moduli of rational maps with prescribed singularity data via Grassmannians of subspaces of $x\in X$01. Connections to Cremona groups (where linear-type elements generate the planar Cremona group), Rees algebras and integral closure, Donaldson–Thomas theory, tropical geometry, birational dynamics, and positive characteristic are indicated but not developed.

Conclusion

The paper establishes that singularities of rational maps admit a substantive birational theory: intrinsic invariants ($x\in X$02, $x\in X$03, $x\in X$04) that measure genuinely different phenomena, sharp numerical inequalities with ACC behaviour, boundedness of normalised graphs, and a classification of the mildest singularities on surfaces that connects in an unexpected way to smooth curves, local class groups, and complement theory. The framework reduces to classical objects—base ideals, Rees valuations, polar multiplicities—at smooth points while extending them intrinsically to singular sources, and the surface case provides both a template and a boundary for what is currently provable.