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SESHADRI CONSTANTS ON ARBITRARY POLARIZED SURFACES
A complete, student-friendly account
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ABOUT THIS DOCUMENT
This rewrite turns a research paper into a guided tour. It assumes only
high school algebra, a little calculus, and basic complex numbers. Every
technical word is explained the first time it is used, and the glossary
at the end collects them. Nothing important is skipped: each step of
the proof is stated and justified, so the reader can follow the whole
argument from the question to the answer.
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1. THE QUESTION IN ONE SENTENCE
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Take a smooth complex surface and a positive ("ample") line bundle on
it. Pick r points in general position. Ask: how much of the positivity
of the line bundle survives after we force curves to pass through all r
points with as much multiplicity as possible?
Write H for the self-intersection number L^2. The surprising answer is
that, once r is large enough (how large depends on the surface and the
line bundle but not on the points), the fraction that survives is
exactly the theoretical maximum sqrt(H/r). That is the theorem proved
here.
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2. THE CAST OF CHARACTERS
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SURFACES. A surface means a smooth connected complex projective surface
S. Think of the plane P^2, or the product P^1 x P^1, or any smooth
surface inside a projective space. It has two complex dimensions, so four
real dimensions. A curve is a one-complex-dimensional piece of S.
LINE BUNDLES AND AMPLENESS. A line bundle L on S is the algebraic way to
talk about a family of curves whose local equations we are allowed to
multiply and twist. When L is ample, some positive power kL gives enough
sections to embed S into a projective space; positivity is abundant.
The self-intersection L.L = L^2 is a positive integer that measures
the "size" of L.
SECTIONS AND DIVISORS. A section of L is a global algebraic function
that, in each local trivialization, is a holomorphic function. Its zero
set is an effective divisor, i.e., a curve (possibly reducible) counted
with multiplicities. Conversely every curve arises this way from some
line bundle. So "curve" and "divisor" and "zero locus of a section" are
three views of the same object.
INTERSECTION NUMBERS. If C is a curve on S and L a line bundle, L.C is
the degree of L restricted to C: the number of zeros of a generic
section of L restricted to C, counted correctly. It is a nonnegative
integer, and it vanishes only when that restriction is trivial. In
particular, if D and V are two curves meeting in finitely many points,
D.V counts their intersections with multiplicity.
MULTIPLICITY AT A POINT. If a curve C passes through a point p, its
multiplicity mult_p C is 1 when it crosses transversally (two smooth
branches with distinct tangents), 2 at an ordinary node, larger when the
curve is more tangled at p. Algebraically, in local coordinates (x,z),
a defining function f has mult_p C = m when f vanishes to order m at p,
i.e., all derivatives of total degree < m vanish and some of degree m do
not.
BLOW-UP. The blow-up of S at a point p replaces p with a curve E of
directions, called the exceptional curve; a curve through p of
multiplicity m becomes a curve meeting E in m points (counted with
multiplicity). Blowing up r points produces pi : Y -> S with exceptional
curves E_1,...,E_r. This is a standard way to record multiplicities as
intersection numbers.
VERY GENERAL POINTS. A property P of r-tuples of distinct points holds
at very general points if the set of tuples where it fails is contained
in a countable union of proper Zariski-closed subsets of
U_r = {(p_1,...,p_r) : p_i distinct}. A Zariski-closed subset is one
defined by polynomial equations; if it is proper, its complement is
"generic". Intuitively: the property holds "almost everywhere, in every
way we can reasonably test".
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3. THE SESHADRI CONSTANT
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The multipoint Seshadri constant is
eps(S,L;p) = inf_C (L.C) / (sum_i mult_{p_i} C),
the infimum taken over all integral curves C that meet at least one of
the points. The numerator is how big the curve is; the denominator is
how much of it is concentrated at the points. A curve with huge
multiplicity at the points but tiny degree is exactly what makes the
ratio small.
Equivalently, on the blow-up pi : Y -> S,
eps(S,L;p) = the largest lambda >= 0 such that
pi^*L - lambda (E_1 + ... + E_r) is nef.
"nef" means "nonnegative on every integral curve": a real divisor class
D is nef if D.C >= 0 for every integral curve C. Nefness of the
displayed class says: after subtracting lambda copies of every
exceptional curve, we still meet every curve nonnegatively. That is the
same as eps, by the dictionary between curves through the points and
their strict transforms on Y.
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4. THE UNIVERSAL UPPER BOUND
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For every smooth surface S, every ample L, every r distinct points,
eps(S,L;p) <= sqrt(L^2 / r).
The proof is a dimension count, and it is worth doing carefully because
the same count is the engine of the whole subject.
Let H = L^2. Fix k and put m = ceil(k alpha) for some 0 < alpha. The
sections of kL form a vector space of dimension
h^0(S,kL) = (H/2) k^2 + O(k)
by the Riemann-Roch theorem (asymptotically; the error is linear in k
because the fixed surface contributes bounded terms). A section that
vanishes to order at least m at each of the r points imposes, at each
point, all Taylor coefficients of total degree < m. In two variables
there are
binom(m+1,2) = m(m+1)/2
such coefficients, so all r points together impose at most
r * m(m+1)/2 = (r alpha^2 / 2) k^2 + O(k)
conditions. If r alpha^2 < H, then for all large k the number of
conditions is strictly less than the dimension, so a nonzero section of
kL must vanish to order at least m at every p_i. Equivalently, there is
an effective divisor of class k(pi^*L - alpha E) with E = sum E_i.
Nefness of pi^*L - lambda E at the actual value of lambda then forces
lambda <= H/(r alpha). Letting alpha approach sqrt(H/r) gives
eps <= sqrt(H/r).
A tiny example. On P^2 with L = O(1), H = 1. Sections of O(k) have
dimension binom(k+2,2) ~ k^2/2, and one point imposes ~ m^2/2
conditions. So eps(P^2, O(1); one point) <= 1, and indeed it equals 1.
For r points the bound reads eps <= 1/sqrt(r), the plane form of
Nagata's conjecture.
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5. THE MAIN THEOREM
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Theorem. Let S be a smooth integral complex projective surface and L an
ample line bundle on S. There exists an integer r_0 = r_0(S,L) such that
for every integer r >= r_0 the following holds. There is a countable
union Z_r of proper Zariski-closed subsets of U_r, with nonempty
complement, such that for every tuple (p_1,...,p_r) in U_r minus Z_r
the class
pi^*L - sqrt(L^2/r) (E_1 + ... + E_r)
is nef. Consequently
eps(S,L;p) = sqrt(L^2/r) for every such very general tuple.
In words: for every polarized surface, eventually at very general
points, the Seshadri constant is as large as the upper bound permits.
This is the qualitative Nagata-Biran conjecture for surfaces.
Two warnings. First, r_0 depends on S and L but not on the points.
Second, the theorem is "eventual": it says nothing about small r.
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6. A LITTLE HISTORY
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Nagata (1959) introduced the plane problem while studying Hilbert's
fourteenth problem and proved a strict multiplicity inequality when the
number of points is a square at least 16. The qualitative form, asking
only for eventual equality, is due to Szemberg and Szydlek. Biran (1999)
proved packing stability in symplectic geometry, which is the symplectic
shadow of the same phenomenon. Later work by Harbourne, Roe, Roe-Ross,
and others established partial results. The present theorem settles the
qualitative form for arbitrary polarized surfaces.
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7. THE PLAN OF THE PROOF
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The proof is a chain of reductions. Each turn replaces one hard problem
with an easier one until only elementary algebra is left.
Step 1. Jets. A jet test with injective "jet map" says no nonzero
section hides to high order at all points. We show a test for the
leading Taylor monomials lifts to a test for the actual sections.
Step 2. A nodal curve. One integral curve V in a high multiple of L
with an ordinary node at a point p gives an intersection bound that
traps the leading Taylor exponents of every section of kL inside a
quadrilateral kQ_t.
Step 3. Compression. Change torus coordinates and squash the
quadrilateral's slices onto a triangle kP_t of area k^2 H/2 without
changing slice lengths. Tests for the triangle lift to tests for the
quadrilateral.
Step 4. Finite subgroups. For each large r pick a primitive lattice
direction and a subgroup G of the complex torus with exactly r
points. Averaging over G breaks a polynomial into components, each
living in a triangle.
Step 5. Triangle multiplicity. A triangle of area rho^2/2 and vertical
range rho bounds the order of vanishing at the origin of any Laurent
polynomial supported in it by rho.
Step 6. Number theory. For every large r, tune the triangle's shape
so that the determinant spread along a primitive direction equals
sqrt(rH) exactly.
Step 7. Assemble. The preceding steps produce, for each pair (k,m)
with m > k sqrt(H/r), an injective jet test at some r-tuple of
surface points.
Step 8. Very general points. Injectivity is a Zariski-open condition.
Countably many open conditions intersect nontrivially by Baire's
theorem, so very general tuples satisfy all of them at once.
Step 9. Nefness. If a curve violated the desired nef class, a small
rational perturbation plus Riemann-Roch would build a section
contradicting the jet test. Hence the class is nef and eps is maximal.
That is the whole plan. The following sections fill it in.
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8. STEP 1. JETS ARE A LINEAR ALGEBRA PROBLEM
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ORDER OF VANISHING. For a holomorphic function f near a point q of a
smooth surface, ord_q f is the smallest total degree of a nonzero term
in the Taylor expansion of f at q. By convention ord_q 0 = infinity.
For a section of kL, choose a local frame; a change of frame multiplies
f by a holomorphic unit, so ord_q f does not change.
JET TESTS. Let F be a finite-dimensional space of holomorphic germs
near points q_1,...,q_r. The order-m jet map is
J^m_q : F -> O_{q_1}/m_{q_1}^m + ... + O_{q_r}/m_{q_r}^m,
f |-> (f mod m_{q_j}^m)_{j=1}^r,
where O_{q_j} is the ring of holomorphic germs at q_j and m_{q_j} its
maximal ideal. The order-m jet test is injective if J^m_q is injective.
In coordinates, injectivity means the jet matrix, whose columns are the
Taylor coefficients of a basis up to total degree m-1, has full column
rank. Equivalently, no nonzero f in F vanishes to order at least m at
every q_j. The target has r binom(m+1,2) rows.
PERSISTENCE OF A JET TEST (the key lemma). Suppose a family of
holomorphic functions g_{i,s}, 1 <= i <= N, converges together with all
derivatives up to order m-1 to g_{i,0} at each q_j as s -> 0. If the
limits g_{1,0},...,g_{N,0} have an injective order-m jet test, then so do
g_{1,s},...,g_{N,s} for all small positive s. Proof: a nonzero N x N
minor of the jet matrix at s = 0 has entries that converge to its limit
entries, so it stays nonzero for small s.
INITIAL EXPONENTS. Fix t > 0. On exponent pairs (alpha,beta) in
Z_{>=0}^2 impose the lexicographic order by (alpha + t beta, beta).
Every nonzero power series f in C{x,z} has a first nonzero term in this
order; call its exponent nu_t(f). For a finite-dimensional space F of
power series, the set E_t(F) = {nu_t(f) : 0 != f in F} has exactly dim F
elements, and F has a basis whose initial exponents are these distinct
elements. (Greedily pick first a nonzero functional on F, then continue
on its kernel.)
TRANSFER FROM INITIAL MONOMIALS (Proposition). Let F be a finite-
dimensional space of convergent power series at the origin and t > 0.
Suppose the monomial space spanned by X^alpha Z^beta for (alpha,beta) in
E_t(F) has an injective order-m jet test at r distinct torus points.
Then representatives of F on a neighborhood of the origin have an
injective order-m jet test at r distinct points arbitrarily close to the
origin.
Sketch of proof (the dilation trick).
(a) Choose a basis f_1,...,f_N with distinct initial exponents
e_i = (alpha_i, beta_i), each with initial coefficient 1.
(b) Separate initial terms by a single weight. Put t' = t + eta for a
small eta > 0. For each i, e_i becomes the unique term of least weight
alpha + t' beta in f_i. (If a competing term had equal old weight, its
beta is strictly larger, so raising t helps; if its old weight was
strictly larger, that persists for small eta; terms with old weight at
least M stay above M because beta >= 0.) Let lambda_i = alpha_i + t'beta_i.
(c) Rescale:
g_{i,s}(X,Z) = s^{-lambda_i} f_i(sX, s^{t'}Z).
As s -> 0 and uniformly on bounded sets, g_{i,s} converges to
X^{alpha_i} Z^{beta_i}. The uniform majorant is summable because
alpha + t'beta - lambda_i >= 0 with equality only at e_i, and the polydisc
of absolute convergence is bounded. Derivatives converge too, by Cauchy's
integral formula on slightly larger polydiscs.
(d) The monomial limits have an injective jet test at the given torus
points, so persistence gives the same conclusion for the g_{i,s} for all
small positive s.
(e) The map D_s(X,Z) = (sX, s^{t'}Z) is invertible; pulling back from
D_s(q_j) gives the conclusion for the originals f_i at the points p_j =
D_s(q_j), which are distinct and approach the origin. Done.
Two comments. The auxiliary weight t' only separates finitely many
initial terms; it does not change their exponents. Nothing here requires
t or eta to be rational.
HOW THIS HELPS. For global sections of kL, restrict to a local chart and
a local frame. Restriction is injective because a nonzero section that
vanishes on an open set vanishes everywhere (identity principle on a
connected surface). A frame change multiplies each germ by a unit whose
nonzero constant term does not change the initial exponent. So if all
the initial exponents of H^0(S,kL) lie in a bounded set Omega, any
injective jet test for the larger monomial space M(Omega) transfers back
to the surface.
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9. STEP 2. A NODAL CURVE AND A QUADRILATERAL
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NODAL DIVISOR LEMMA. There exist a positive integer d, an integral
divisor V in |dL|, and a point p in V, such that V is smooth away from p
and has two smooth transverse branches at p.
Proof idea. Pick p and choose e > 0 with eL very ample; embed S so that
p = [1:0:...:0]. Blow up p, getting pi : X -> S with exceptional curve
E. The blow-up sits in S x P^{N-1}, and O(1,1) restricts to
A = O_X(2e pi^*L - E). Then A^2 = O_X(4e pi^*L - 2E) is very ample.
Bertini makes a general member Vtilde of this linear system smooth and
integral, and it can be arranged to meet E transversely in two distinct
points (because the restriction of A^2 to E is O(2), whose sections
separate two points). Push down: V = pi(Vtilde) lies in |4eL|, is
integral, is smooth away from p, and has two smooth branches at p with
distinct tangents. Set d = 4e.
COORDINATES. Choose holomorphic coordinates (x,z) at p whose coordinate
axes are the two branches. In a local frame of L, the defining section
tau of V has the form
tau = u(x,z) * x * z, with u(0,0) != 0.
DEMAND ON t. Choose T > 1 so large that
(1+t)^2 > d^2 H t for every t > T.
This is possible because the left side is quadratic in t and the right
side is linear.
THE QUADRILATERAL. For t > T put
c = 1/d, W = dH, a = W t/(1+t), b = W/(1+t),
Q_t = conv{ (0,0), (a,0), (c,c), (0,b) }.
The inequality is equivalent to c/a + c/b > 1, which says that (c,c)
lies beyond the segment joining (a,0) to (0,b), on the far side from the
origin. So the vertices of Q_t appear in the displayed cyclic order.
Also define the residual triangle
Delta_t = conv{ (0,0), (a,0), (0,b) }.
QUADRILATERAL BOUND (Proposition). For every k >= 1, every t > T, and
every nonzero s in H^0(S,kL),
nu_t(s) lies in k Q_t.
Moreover H^0(S,kL) has a basis with pairwise distinct initial exponents,
all lying in k Q_t.
Proof. Let D = div(s). First suppose V is not contained in D, so the
local function f of s is nonzero on each branch. Claim: neither f(x,0)
nor f(0,z) is identically zero. If f(x,0) = 0, the whole branch {z = 0}
lies inside D; but D ∩ V is a proper closed subset of the integral curve
V, hence finite, and an entire branch cannot fit in a finite set. The
same argument handles the other branch.
Let l_x = ord_0 f(x,0) and l_z = ord_0 f(0,z), and let v be the weight
of nu_t(f). The pure term giving l_x has weight l_x, so l_x >= v; the
pure term giving l_z has weight t l_z, so l_z >= v/t. Intersection
numbers add over the two smooth branches at p, so
l_x + l_z = I_p(D,V) <= D.V = k d H.
Therefore v(1 + 1/t) <= k d H. Since ka = k W t/(1+t) = k d H t/(1+t),
and (1 + 1/t) = (1+t)/t, this gives v <= ka. So nu_t(s) lies in
k Delta_t, which is a subset of k Q_t.
For a general nonzero s, write s = tau^j rho with j the coefficient of V
in div(s), rho in H^0(S, nL), n = k - jd, and R = div(rho) not
containing V. Ampleness gives R ∩ V finite, so the preceding argument
applies to rho. Orders add under multiplication of germs, and the unit u
contributes exponent (0,0), so
nu_t(s) = (j,j) + nu_t(rho).
Since nu_t(rho)/n lies in Delta_t,
nu_t(s)/k = (n/k) (nu_t(rho)/n) + (jd/k) (c,c).
The two nonnegative coefficients sum to 1, so this convex combination
lies in Q_t. If n = 0, then rho is a nonzero constant (R = 0) and
nu_t(s) = (j,j) = k(c,c), the vertex of kQ_t.
Finally, restriction to the analytic chart is injective on global
sections. Apply the basis-with-distinct-initial-exponents statement (the
finite coefficient-elimination argument) to this finite-dimensional
space of germs: it produces a basis of H^0(S,kL) with distinct initial
exponents, each satisfying the bound just proved.
Interpretation. Q_t remembers exactly two facts: the intersection bound
D.V <= kdH, and the fact that the two branches at p are smooth and
transverse. The parameter t controls the trade-off between the two
branches. As t grows, a approaches dH and b shrinks toward 0.