id	sid	tid	token	lemma	pos
nz805x2475b	1	1	this	this	DET
nz805x2475b	1	2	dissertation	dissertation	NOUN
nz805x2475b	1	3	examines	examine	VERB
nz805x2475b	1	4	two	two	NUM
nz805x2475b	1	5	related	related	ADJ
nz805x2475b	1	6	topics	topic	NOUN
nz805x2475b	1	7	,	,	PUNCT
nz805x2475b	1	8	defining	define	VERB
nz805x2475b	1	9	equations	equation	NOUN
nz805x2475b	1	10	of	of	ADP
nz805x2475b	1	11	rees	ree	NOUN
nz805x2475b	1	12	algebras	algebra	NOUN
nz805x2475b	1	13	in	in	ADP
nz805x2475b	1	14	two	two	NUM
nz805x2475b	1	15	variables	variable	NOUN
nz805x2475b	1	16	and	and	CCONJ
nz805x2475b	1	17	the	the	DET
nz805x2475b	1	18	geometry	geometry	NOUN
nz805x2475b	1	19	of	of	ADP
nz805x2475b	1	20	rational	rational	ADJ
nz805x2475b	1	21	plane	plane	NOUN
nz805x2475b	1	22	curves	curve	NOUN
nz805x2475b	1	23	.	.	PUNCT
nz805x2475b	2	1	first	first	ADV
nz805x2475b	2	2	,	,	PUNCT
nz805x2475b	2	3	we	we	PRON
nz805x2475b	2	4	consider	consider	VERB
nz805x2475b	2	5	the	the	DET
nz805x2475b	2	6	ideal	ideal	ADJ
nz805x2475b	2	7	k	k	X
nz805x2475b	2	8	of	of	ADP
nz805x2475b	2	9	defining	define	VERB
nz805x2475b	2	10	equations	equation	NOUN
nz805x2475b	2	11	of	of	ADP
nz805x2475b	2	12	the	the	DET
nz805x2475b	2	13	rees	ree	NOUN
nz805x2475b	2	14	algebra	algebra	NOUN
nz805x2475b	2	15	of	of	ADP
nz805x2475b	2	16	an	an	DET
nz805x2475b	2	17	ideal	ideal	NOUN
nz805x2475b	2	18	i	i	PRON
nz805x2475b	2	19	in	in	ADP
nz805x2475b	2	20	the	the	DET
nz805x2475b	2	21	polynomial	polynomial	ADJ
nz805x2475b	2	22	ring	ring	NOUN
nz805x2475b	2	23	k[x_1,x_2	k[x_1,x_2	NOUN
nz805x2475b	2	24	]	]	PUNCT
nz805x2475b	2	25	generated	generate	VERB
nz805x2475b	2	26	by	by	ADP
nz805x2475b	2	27	forms	form	NOUN
nz805x2475b	2	28	of	of	ADP
nz805x2475b	2	29	the	the	DET
nz805x2475b	2	30	same	same	ADJ
nz805x2475b	2	31	degree	degree	NOUN
nz805x2475b	2	32	.	.	PUNCT
nz805x2475b	3	1	we	we	PRON
nz805x2475b	3	2	give	give	VERB
nz805x2475b	3	3	a	a	DET
nz805x2475b	3	4	description	description	NOUN
nz805x2475b	3	5	of	of	ADP
nz805x2475b	3	6	the	the	DET
nz805x2475b	3	7	part	part	NOUN
nz805x2475b	3	8	of	of	ADP
nz805x2475b	3	9	k	k	PROPN
nz805x2475b	3	10	in	in	ADP
nz805x2475b	3	11	degree	degree	NOUN
nz805x2475b	3	12	greater	great	ADJ
nz805x2475b	3	13	than	than	ADP
nz805x2475b	3	14	or	or	CCONJ
nz805x2475b	3	15	equal	equal	ADJ
nz805x2475b	3	16	to	to	ADP
nz805x2475b	3	17	the	the	DET
nz805x2475b	3	18	second	second	ADV
nz805x2475b	3	19	largest	large	ADJ
nz805x2475b	3	20	degree	degree	NOUN
nz805x2475b	3	21	of	of	ADP
nz805x2475b	3	22	a	a	DET
nz805x2475b	3	23	generator	generator	NOUN
nz805x2475b	3	24	of	of	ADP
nz805x2475b	3	25	the	the	DET
nz805x2475b	3	26	syzygy	syzygy	PROPN
nz805x2475b	3	27	module	module	NOUN
nz805x2475b	3	28	of	of	ADP
nz805x2475b	3	29	i.	i.	PROPN
nz805x2475b	3	30	this	this	DET
nz805x2475b	3	31	description	description	NOUN
nz805x2475b	3	32	lets	let	VERB
nz805x2475b	3	33	us	we	PRON
nz805x2475b	3	34	compute	compute	VERB
nz805x2475b	3	35	the	the	DET
nz805x2475b	3	36	minimal	minimal	ADJ
nz805x2475b	3	37	generators	generator	NOUN
nz805x2475b	3	38	of	of	ADP
nz805x2475b	3	39	k	k	PROPN
nz805x2475b	3	40	in	in	ADP
nz805x2475b	3	41	these	these	DET
nz805x2475b	3	42	degrees	degree	NOUN
nz805x2475b	3	43	.	.	PUNCT
nz805x2475b	4	1	when	when	SCONJ
nz805x2475b	4	2	i	i	PRON
nz805x2475b	4	3	is	be	AUX
nz805x2475b	4	4	generated	generate	VERB
nz805x2475b	4	5	by	by	ADP
nz805x2475b	4	6	three	three	NUM
nz805x2475b	4	7	forms	form	NOUN
nz805x2475b	4	8	,	,	PUNCT
nz805x2475b	4	9	we	we	PRON
nz805x2475b	4	10	also	also	ADV
nz805x2475b	4	11	produce	produce	VERB
nz805x2475b	4	12	a	a	DET
nz805x2475b	4	13	generating	generating	NOUN
nz805x2475b	4	14	set	set	NOUN
nz805x2475b	4	15	for	for	ADP
nz805x2475b	4	16	each	each	DET
nz805x2475b	4	17	graded	grade	VERB
nz805x2475b	4	18	slice	slice	NOUN
nz805x2475b	4	19	k_i	k_i	PUNCT
nz805x2475b	4	20	in	in	ADP
nz805x2475b	4	21	this	this	PRON
nz805x2475b	4	22	range.the	range.the	DET
nz805x2475b	4	23	second	second	ADJ
nz805x2475b	4	24	topic	topic	NOUN
nz805x2475b	4	25	we	we	PRON
nz805x2475b	4	26	study	study	VERB
nz805x2475b	4	27	is	be	AUX
nz805x2475b	4	28	the	the	DET
nz805x2475b	4	29	geometry	geometry	NOUN
nz805x2475b	4	30	of	of	ADP
nz805x2475b	4	31	birationally	birationally	ADV
nz805x2475b	4	32	parametrized	parametrize	VERB
nz805x2475b	4	33	plane	plane	NOUN
nz805x2475b	4	34	curves	curve	VERB
nz805x2475b	4	35	c	c	NOUN
nz805x2475b	4	36	,	,	PUNCT
nz805x2475b	4	37	particularly	particularly	ADV
nz805x2475b	4	38	their	their	PRON
nz805x2475b	4	39	singularities	singularity	NOUN
nz805x2475b	4	40	.	.	PUNCT
nz805x2475b	5	1	this	this	PRON
nz805x2475b	5	2	is	be	AUX
nz805x2475b	5	3	connected	connect	VERB
nz805x2475b	5	4	to	to	ADP
nz805x2475b	5	5	the	the	DET
nz805x2475b	5	6	first	first	ADJ
nz805x2475b	5	7	part	part	NOUN
nz805x2475b	5	8	because	because	SCONJ
nz805x2475b	5	9	the	the	DET
nz805x2475b	5	10	parametric	parametric	ADJ
nz805x2475b	5	11	equations	equation	NOUN
nz805x2475b	5	12	generate	generate	VERB
nz805x2475b	5	13	an	an	DET
nz805x2475b	5	14	ideal	ideal	NOUN
nz805x2475b	5	15	i	i	PRON
nz805x2475b	5	16	in	in	ADP
nz805x2475b	5	17	k[x_1,x_2	k[x_1,x_2	NOUN
nz805x2475b	5	18	]	]	PUNCT
nz805x2475b	5	19	,	,	PUNCT
nz805x2475b	5	20	and	and	CCONJ
nz805x2475b	5	21	the	the	DET
nz805x2475b	5	22	rees	ree	NOUN
nz805x2475b	5	23	algebra	algebra	NOUN
nz805x2475b	5	24	of	of	ADP
nz805x2475b	5	25	i	i	PRON
nz805x2475b	5	26	is	be	AUX
nz805x2475b	5	27	the	the	DET
nz805x2475b	5	28	coordinate	coordinate	NOUN
nz805x2475b	5	29	ring	ring	NOUN
nz805x2475b	5	30	of	of	ADP
nz805x2475b	5	31	the	the	DET
nz805x2475b	5	32	graph	graph	NOUN
nz805x2475b	5	33	of	of	ADP
nz805x2475b	5	34	the	the	DET
nz805x2475b	5	35	parametrization	parametrization	NOUN
nz805x2475b	5	36	.	.	PUNCT
nz805x2475b	6	1	we	we	PRON
nz805x2475b	6	2	show	show	VERB
nz805x2475b	6	3	that	that	SCONJ
nz805x2475b	6	4	the	the	DET
nz805x2475b	6	5	fitting	fitting	ADJ
nz805x2475b	6	6	ideals	ideal	NOUN
nz805x2475b	6	7	of	of	ADP
nz805x2475b	6	8	a	a	DET
nz805x2475b	6	9	slice	slice	NOUN
nz805x2475b	6	10	of	of	ADP
nz805x2475b	6	11	the	the	DET
nz805x2475b	6	12	symmetric	symmetric	ADJ
nz805x2475b	6	13	algebra	algebra	NOUN
nz805x2475b	6	14	of	of	ADP
nz805x2475b	6	15	i	i	PRON
nz805x2475b	6	16	detect	detect	VERB
nz805x2475b	6	17	the	the	DET
nz805x2475b	6	18	singular	singular	ADJ
nz805x2475b	6	19	points	point	NOUN
nz805x2475b	6	20	on	on	ADP
nz805x2475b	6	21	or	or	CCONJ
nz805x2475b	6	22	infinitely	infinitely	ADV
nz805x2475b	6	23	near	near	ADV
nz805x2475b	6	24	to	to	ADP
nz805x2475b	6	25	c	c	PROPN
nz805x2475b	6	26	as	as	ADV
nz805x2475b	6	27	well	well	ADV
nz805x2475b	6	28	as	as	ADP
nz805x2475b	6	29	their	their	PRON
nz805x2475b	6	30	multiplicities	multiplicity	NOUN
nz805x2475b	6	31	.	.	PUNCT
nz805x2475b	7	1	using	use	VERB
nz805x2475b	7	2	this	this	PRON
nz805x2475b	7	3	,	,	PUNCT
nz805x2475b	7	4	we	we	PRON
nz805x2475b	7	5	stratify	stratify	VERB
nz805x2475b	7	6	the	the	DET
nz805x2475b	7	7	space	space	NOUN
nz805x2475b	7	8	of	of	ADP
nz805x2475b	7	9	hilbert	hilbert	PROPN
nz805x2475b	7	10	-	-	PUNCT
nz805x2475b	7	11	burch	burch	NOUN
nz805x2475b	7	12	matrices	matrix	NOUN
nz805x2475b	7	13	of	of	ADP
nz805x2475b	7	14	parametrizations	parametrization	NOUN
nz805x2475b	7	15	into	into	ADP
nz805x2475b	7	16	locally	locally	ADV
nz805x2475b	7	17	closed	close	VERB
nz805x2475b	7	18	subsets	subset	NOUN
nz805x2475b	7	19	representing	represent	VERB
nz805x2475b	7	20	those	those	DET
nz805x2475b	7	21	curves	curve	NOUN
nz805x2475b	7	22	with	with	ADP
nz805x2475b	7	23	a	a	DET
nz805x2475b	7	24	given	give	VERB
nz805x2475b	7	25	number	number	NOUN
nz805x2475b	7	26	of	of	ADP
nz805x2475b	7	27	points	point	NOUN
nz805x2475b	7	28	of	of	ADP
nz805x2475b	7	29	each	each	DET
nz805x2475b	7	30	multiplicity	multiplicity	NOUN
nz805x2475b	7	31	.	.	PUNCT
nz805x2475b	8	1	finally	finally	ADV
nz805x2475b	8	2	,	,	PUNCT
nz805x2475b	8	3	we	we	PRON
nz805x2475b	8	4	connect	connect	VERB
nz805x2475b	8	5	one	one	NUM
nz805x2475b	8	6	of	of	ADP
nz805x2475b	8	7	these	these	DET
nz805x2475b	8	8	fitting	fitting	ADJ
nz805x2475b	8	9	ideals	ideal	NOUN
nz805x2475b	8	10	to	to	ADP
nz805x2475b	8	11	a	a	DET
nz805x2475b	8	12	slice	slice	NOUN
nz805x2475b	8	13	of	of	ADP
nz805x2475b	8	14	the	the	DET
nz805x2475b	8	15	ideal	ideal	ADJ
nz805x2475b	8	16	k.	k.	NOUN
nz805x2475b	8	17	in	in	ADP
nz805x2475b	8	18	particular	particular	ADJ
nz805x2475b	8	19	,	,	PUNCT
nz805x2475b	8	20	we	we	PRON
nz805x2475b	8	21	give	give	VERB
nz805x2475b	8	22	a	a	DET
nz805x2475b	8	23	complete	complete	ADJ
nz805x2475b	8	24	list	list	NOUN
nz805x2475b	8	25	of	of	ADP
nz805x2475b	8	26	the	the	DET
nz805x2475b	8	27	possible	possible	ADJ
nz805x2475b	8	28	bidegrees	bidegree	NOUN
nz805x2475b	8	29	of	of	ADP
nz805x2475b	8	30	a	a	DET
nz805x2475b	8	31	minimal	minimal	ADJ
nz805x2475b	8	32	generating	generating	NOUN
nz805x2475b	8	33	set	set	NOUN
nz805x2475b	8	34	of	of	ADP
nz805x2475b	8	35	k	k	PROPN
nz805x2475b	8	36	when	when	SCONJ
nz805x2475b	8	37	the	the	DET
nz805x2475b	8	38	degree	degree	NOUN
nz805x2475b	8	39	of	of	ADP
nz805x2475b	8	40	c	c	PROPN
nz805x2475b	8	41	is	be	AUX
nz805x2475b	8	42	7	7	NUM
nz805x2475b	8	43	,	,	PUNCT
nz805x2475b	8	44	as	as	ADV
nz805x2475b	8	45	well	well	ADV
nz805x2475b	8	46	as	as	ADP
nz805x2475b	8	47	the	the	DET
nz805x2475b	8	48	correspondence	correspondence	NOUN
nz805x2475b	8	49	between	between	ADP
nz805x2475b	8	50	these	these	DET
nz805x2475b	8	51	bidegrees	bidegree	NOUN
nz805x2475b	8	52	and	and	CCONJ
nz805x2475b	8	53	the	the	DET
nz805x2475b	8	54	multiplicities	multiplicity	NOUN
nz805x2475b	8	55	of	of	ADP
nz805x2475b	8	56	the	the	DET
nz805x2475b	8	57	points	point	NOUN
nz805x2475b	8	58	of	of	ADP
nz805x2475b	8	59	c	c	PROPN
nz805x2475b	8	60	,	,	PUNCT
nz805x2475b	8	61	which	which	PRON
nz805x2475b	8	62	was	be	AUX
nz805x2475b	8	63	previously	previously	ADV
nz805x2475b	8	64	known	know	VERB
nz805x2475b	8	65	only	only	ADV
nz805x2475b	8	66	for	for	ADP
nz805x2475b	8	67	degree	degree	NOUN
nz805x2475b	8	68	&	&	CCONJ
nz805x2475b	8	69	lt;=	lt;=	ADP
nz805x2475b	8	70	6	6	NUM
nz805x2475b	8	71	.	.	PUNCT