Abstract
A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen–Macaulay projective scheme is connected. We give a quantitative version: If (Formula presented.) is an arithmetically Gorenstein projective scheme of regularity (Formula presented.), and if every irreducible component of X has regularity (Formula presented.), we show that the dual graph of X is (Formula presented.)-connected. The bound is sharp. We also provide a strong converse to Hartshorne’s result: every connected graph is dual to a suitable arithmetically Cohen–Macaulay projective curve of regularity (Formula presented.)3, whose components are all rational normal curves. The regularity bound is smallest possible in general. Further consequences are: (1) every graph is the Hochster–Huneke graph of a complete equidimensional local ring. (This answers a question by Sather–Wagstaff and Spiroff). (2) The regularity of a curve is not larger than the sum of the regularities of its primary components.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1-18 |
| Number of pages | 18 |
| Journal | Journal of Algebraic Combinatorics |
| DOIs | |
| State | Accepted/In press - Feb 27 2017 |
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Keywords
- Arithmetically Gorenstein arrangements
- Balinski’s theorem
- Castelnuovo–Mumford regularity
- Dual graphs
- k-connectivity
ASJC Scopus subject areas
- Algebra and Number Theory
- Discrete Mathematics and Combinatorics
Cite this
Regulating Hartshorne’s connectedness theorem. / Benedetti, Bruno; Bolognese, Barbara; Varbaro, Matteo.
In: Journal of Algebraic Combinatorics, 27.02.2017, p. 1-18.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Regulating Hartshorne’s connectedness theorem
AU - Benedetti, Bruno
AU - Bolognese, Barbara
AU - Varbaro, Matteo
PY - 2017/2/27
Y1 - 2017/2/27
N2 - A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen–Macaulay projective scheme is connected. We give a quantitative version: If (Formula presented.) is an arithmetically Gorenstein projective scheme of regularity (Formula presented.), and if every irreducible component of X has regularity (Formula presented.), we show that the dual graph of X is (Formula presented.)-connected. The bound is sharp. We also provide a strong converse to Hartshorne’s result: every connected graph is dual to a suitable arithmetically Cohen–Macaulay projective curve of regularity (Formula presented.)3, whose components are all rational normal curves. The regularity bound is smallest possible in general. Further consequences are: (1) every graph is the Hochster–Huneke graph of a complete equidimensional local ring. (This answers a question by Sather–Wagstaff and Spiroff). (2) The regularity of a curve is not larger than the sum of the regularities of its primary components.
AB - A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen–Macaulay projective scheme is connected. We give a quantitative version: If (Formula presented.) is an arithmetically Gorenstein projective scheme of regularity (Formula presented.), and if every irreducible component of X has regularity (Formula presented.), we show that the dual graph of X is (Formula presented.)-connected. The bound is sharp. We also provide a strong converse to Hartshorne’s result: every connected graph is dual to a suitable arithmetically Cohen–Macaulay projective curve of regularity (Formula presented.)3, whose components are all rational normal curves. The regularity bound is smallest possible in general. Further consequences are: (1) every graph is the Hochster–Huneke graph of a complete equidimensional local ring. (This answers a question by Sather–Wagstaff and Spiroff). (2) The regularity of a curve is not larger than the sum of the regularities of its primary components.
KW - Arithmetically Gorenstein arrangements
KW - Balinski’s theorem
KW - Castelnuovo–Mumford regularity
KW - Dual graphs
KW - k-connectivity
UR - http://www.scopus.com/inward/record.url?scp=85014048060&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85014048060&partnerID=8YFLogxK
U2 - 10.1007/s10801-017-0744-8
DO - 10.1007/s10801-017-0744-8
M3 - Article
AN - SCOPUS:85014048060
SP - 1
EP - 18
JO - Journal of Algebraic Combinatorics
JF - Journal of Algebraic Combinatorics
SN - 0925-9899
ER -