Open problems on negativity in algebraic
geometry and connections to combinatorics and
commutative algebra
Brian Harbourne
Department of Mathematics
University of Nebraska-Lincoln
Special Session on Commutative Algebra and Its Interactions
with Algebraic Geometry
Organized by Susan Cooper, Sara Faridi, and William Traves
held at Dalhousie University
October 18, 2014
New Title! (same content ...)
Hadean numbers: exploring the underworld of
reduced singular plane curves
Brian Harbourne
Department of Mathematics
University of Nebraska-Lincoln
Special Session on Commutative Algebra and Its Interactions
with Algebraic Geometry
Organized by Susan Cooper, Sara Faridi, and William Traves
held at Dalhousie University
October 18, 2014
This talk is based on:
Bounded Negativity and Arrangements of Lines (arXiv:1407.2966 )
Thomas Bauer,
Sandra Di Rocco,
Jack Huizenga,
Anders Lundman,
Piotr Pokora, and
Tomasz Szemberg.
For simplicity assume char = 0.
(What’s interesting is less clear and more complicated for char > 0.)
Motivation: Bounded Negativity Conjecture
Fix a choice of points p1 , . . . , ps ∈ P2 .
Let C be a reduced plane curve of degree d.
Define QS (C ) = d 2 − multp1 (C )2 − . . . − multps (C )2 .
Open Problem: Is there an integer NS s.t. QS (C ) ≥ NS for all C ?
Old Open Conjecture (Bounded Negativity Conj.): Yes, there is.
This is old, open and hard. Is there a stepping stone problem?
Suppose we vary both C and S?
Underlying Question
New Question: How negative can QS (C ) be?
Let C be a singular plane curve of degree d with singularities of
multiplicities m1 , . . . , ms .
Define Q(C ) = d 2 − m12 − · · · − ms2 .
Examples:
d nodes
2
d
...
degree = d+1
2
2
2
2
Q = (d + 1) − d − d · 2
= −2d + 1
2
Q = (d + 1)2 −
d 2
2
2
= −d 2 + 4d + 1
Which is more negative?
The only fair way to compare is on a pro rata basis . . .
Hadean numbers
Let C be a singular plane curve of degree d with singularities of
multiplicities m1 , . . . , ms .
Define H(C ) =
d 2 − m12 − · · · − ms2
Q(C )
=
.
s
s
Examples:
d nodes
2
d
...
degree = d+1
2
2
−2d + 1
3
H=
= −2 +
d +1
d +1
H=
−d 2 + 4d + 1
d 2 −3d+2
2
= −2 +
d2
2d + 6
− 3d + 2
Open Question: What Hadean numbers occur?
Curves C with H(C ) > −2 are abundant and easy to see.
Curves with H(C ) ≤ −2 form a mysterious underworld still largely
invisible:
No irreducible C yet known with H(C ) ≤ −2.
(If such a C exists, deg(C ) ≥ 21.)
Consider the opposite extreme: curves which are unions of lines.
Linearization
d
2
d lines give
Thus:
d
2
m1
2
=
mi
2
pairs,
+ ··· +
pairs at each singularity of multiplicity mi
ms
2
(hence d 2 −
Example: 4 lines so
4
2
= 6 pairs:
4
2
2
2
+
i.e.,
=
3
2
+
2
2
+
P
mi2 = d −
P
mi )
3 pairs
2
2
1 pair each
Thus
d − m1 − · · · − ms
d 2 − m12 − · · · − ms2
=
s
s
d − 2t2 − 3t3 − · · · − dtd
=
s
H(C ) =
where tk is the number of points of multiplicity k.
Bounds
Melchior’s Theorem: For nonconcurrent lines over the reals
X
t2 ≥ 3 +
(k − 3)tk .
k>3
Proof: χ(RP2 ) = 1
Corollary: If C is a union of real lines, then H(C ) > −3
(and examples show that the infimum is equal to −3).
Similarly (but harder and deeper): If C is a union of complex lines,
then H(C ) > −4, but infimum is unknown.
Examples:
Klein (1879): 21 lines with t4 = 21, t3 = 28: H = −3
Wiman (1890): 45 lines with t5 = 36, t4 = 45, t3 = 120: H = −3.36
Summary: Cosmology of Reduced Singular Plane Curves
H=5
H=
H≥ 0
H = −1
H = −2
H = −3
H ≤ −4
4
3
alla Botticelli’s Mappa dell’inferno
H=5
H=
4
3
H≥ 0
H = −1
H = −2
H = −3
2
H = −2 + No irreducibles yet known below H = −2
H = −2.2
H = −3
Klein H = −3
Wiman H = −3.36
H ≤ −4
?
Nothing yet known with H ≤ −4
Related Problems
Let I = ∩i I (pi ) be the ideal of points p1 , . . . , ps ∈ P2 .
The mth symbolic power is I (m) = ∩i (I (pi )m ).
Theorem (ELS, HH): I (4) ⊆ I 2 .
Problem: Classify all cases of I (3) 6⊆ I 2 .
Observation: Whenever I is the ideal of the singularities of a
union of lines with t2 = 0, then I (3) 6⊆ I 2 seems to hold.
Only three cases of complex line arrangements are known with
t2 = 0.
Are there others? If so, must I (3) 6⊆ I 2 hold for these too?