id	sid	tid	token	lemma	pos
st74cn73540	1	1	in	in	ADP
st74cn73540	1	2	this	this	DET
st74cn73540	1	3	thesis	thesis	NOUN
st74cn73540	1	4	we	we	PRON
st74cn73540	1	5	explore	explore	VERB
st74cn73540	1	6	three	three	NUM
st74cn73540	1	7	projects	project	NOUN
st74cn73540	1	8	,	,	PUNCT
st74cn73540	1	9	all	all	PRON
st74cn73540	1	10	related	relate	VERB
st74cn73540	1	11	to	to	ADP
st74cn73540	1	12	the	the	DET
st74cn73540	1	13	special	special	ADJ
st74cn73540	1	14	fiber	fiber	NOUN
st74cn73540	1	15	ring	ring	NOUN
st74cn73540	1	16	somehow	somehow	ADV
st74cn73540	1	17	.	.	PUNCT
st74cn73540	2	1	consider	consider	VERB
st74cn73540	2	2	a	a	DET
st74cn73540	2	3	rational	rational	ADJ
st74cn73540	2	4	map	map	NOUN
st74cn73540	2	5	from	from	ADP
st74cn73540	2	6	projective	projective	ADJ
st74cn73540	2	7	d-1	d-1	NOUN
st74cn73540	2	8	space	space	NOUN
st74cn73540	2	9	to	to	PART
st74cn73540	2	10	projective	projective	VERB
st74cn73540	2	11	m-1	m-1	NUM
st74cn73540	2	12	space	space	NOUN
st74cn73540	2	13	given	give	VERB
st74cn73540	2	14	by	by	ADP
st74cn73540	2	15	m	m	NOUN
st74cn73540	2	16	many	many	ADJ
st74cn73540	2	17	degree	degree	NOUN
st74cn73540	2	18	δ	δ	PROPN
st74cn73540	2	19	forms	form	NOUN
st74cn73540	2	20	,	,	PUNCT
st74cn73540	2	21	g_1,	g_1,	X
st74cn73540	2	22	...	...	PROPN
st74cn73540	2	23	,g_m	,g_m	PUNCT
st74cn73540	2	24	,	,	PUNCT
st74cn73540	2	25	in	in	ADP
st74cn73540	2	26	d	d	NOUN
st74cn73540	2	27	variables	variable	NOUN
st74cn73540	2	28	.	.	PUNCT
st74cn73540	3	1	let	let	VERB
st74cn73540	3	2	i	i	PRON
st74cn73540	3	3	be	be	AUX
st74cn73540	3	4	the	the	DET
st74cn73540	3	5	ideal	ideal	NOUN
st74cn73540	3	6	generated	generate	VERB
st74cn73540	3	7	by	by	ADP
st74cn73540	3	8	these	these	DET
st74cn73540	3	9	forms	form	NOUN
st74cn73540	3	10	.	.	PUNCT
st74cn73540	4	1	the	the	DET
st74cn73540	4	2	special	special	ADJ
st74cn73540	4	3	fiber	fiber	NOUN
st74cn73540	4	4	ring	ring	NOUN
st74cn73540	4	5	is	be	AUX
st74cn73540	4	6	the	the	DET
st74cn73540	4	7	ring	ring	NOUN
st74cn73540	4	8	corresponding	correspond	VERB
st74cn73540	4	9	to	to	ADP
st74cn73540	4	10	the	the	DET
st74cn73540	4	11	closure	closure	NOUN
st74cn73540	4	12	of	of	ADP
st74cn73540	4	13	the	the	DET
st74cn73540	4	14	image	image	NOUN
st74cn73540	4	15	of	of	ADP
st74cn73540	4	16	this	this	DET
st74cn73540	4	17	map	map	NOUN
st74cn73540	4	18	,	,	PUNCT
st74cn73540	4	19	denoted	denote	VERB
st74cn73540	4	20	f(i).in	f(i).in	ADV
st74cn73540	4	21	the	the	DET
st74cn73540	4	22	first	first	ADJ
st74cn73540	4	23	project	project	NOUN
st74cn73540	4	24	we	we	PRON
st74cn73540	4	25	restrict	restrict	VERB
st74cn73540	4	26	to	to	ADP
st74cn73540	4	27	the	the	DET
st74cn73540	4	28	case	case	NOUN
st74cn73540	4	29	d	d	X
st74cn73540	4	30	=	=	SYM
st74cn73540	4	31	2	2	NUM
st74cn73540	4	32	and	and	CCONJ
st74cn73540	4	33	m	m	VERB
st74cn73540	4	34	=	=	X
st74cn73540	4	35	3	3	NUM
st74cn73540	4	36	.	.	PUNCT
st74cn73540	5	1	in	in	ADP
st74cn73540	5	2	this	this	DET
st74cn73540	5	3	case	case	NOUN
st74cn73540	5	4	proj(f(i	proj(f(i	PROPN
st74cn73540	5	5	)	)	PUNCT
st74cn73540	5	6	)	)	PUNCT
st74cn73540	5	7	is	be	AUX
st74cn73540	5	8	a	a	DET
st74cn73540	5	9	rational	rational	ADJ
st74cn73540	5	10	plane	plane	NOUN
st74cn73540	5	11	curve	curve	NOUN
st74cn73540	5	12	,	,	PUNCT
st74cn73540	5	13	and	and	CCONJ
st74cn73540	5	14	we	we	PRON
st74cn73540	5	15	analyze	analyze	VERB
st74cn73540	5	16	its	its	PRON
st74cn73540	5	17	singularities	singularity	NOUN
st74cn73540	5	18	using	use	VERB
st74cn73540	5	19	the	the	DET
st74cn73540	5	20	syzygy	syzygy	PROPN
st74cn73540	5	21	matrix	matrix	NOUN
st74cn73540	5	22	of	of	ADP
st74cn73540	5	23	i.	i.	PROPN
st74cn73540	5	24	we	we	PRON
st74cn73540	5	25	are	be	AUX
st74cn73540	5	26	specifically	specifically	ADV
st74cn73540	5	27	interested	interested	ADJ
st74cn73540	5	28	in	in	ADP
st74cn73540	5	29	cusps	cusps	NOUN
st74cn73540	5	30	.	.	PUNCT
st74cn73540	6	1	motivated	motivate	VERB
st74cn73540	6	2	by	by	ADP
st74cn73540	6	3	this	this	PRON
st74cn73540	6	4	and	and	CCONJ
st74cn73540	6	5	the	the	DET
st74cn73540	6	6	classical	classical	ADJ
st74cn73540	6	7	plücker	plücker	NOUN
st74cn73540	6	8	relations	relation	NOUN
st74cn73540	6	9	,	,	PUNCT
st74cn73540	6	10	we	we	PRON
st74cn73540	6	11	exhibit	exhibit	VERB
st74cn73540	6	12	a	a	DET
st74cn73540	6	13	curious	curious	ADJ
st74cn73540	6	14	categorization	categorization	NOUN
st74cn73540	6	15	of	of	ADP
st74cn73540	6	16	the	the	DET
st74cn73540	6	17	dual	dual	ADJ
st74cn73540	6	18	curve	curve	NOUN
st74cn73540	6	19	using	use	VERB
st74cn73540	6	20	the	the	DET
st74cn73540	6	21	syzygy	syzygy	PROPN
st74cn73540	6	22	matrix	matrix	NOUN
st74cn73540	6	23	.	.	PUNCT
st74cn73540	7	1	we	we	PRON
st74cn73540	7	2	then	then	ADV
st74cn73540	7	3	use	use	VERB
st74cn73540	7	4	this	this	PRON
st74cn73540	7	5	to	to	PART
st74cn73540	7	6	give	give	VERB
st74cn73540	7	7	previously	previously	ADV
st74cn73540	7	8	unknown	unknown	ADJ
st74cn73540	7	9	bounds	bound	NOUN
st74cn73540	7	10	on	on	ADP
st74cn73540	7	11	the	the	DET
st74cn73540	7	12	number	number	NOUN
st74cn73540	7	13	of	of	ADP
st74cn73540	7	14	cusps	cusps	NOUN
st74cn73540	7	15	for	for	ADP
st74cn73540	7	16	fixed	fix	VERB
st74cn73540	7	17	splitting	splitting	NOUN
st74cn73540	7	18	types.in	types.in	NUM
st74cn73540	7	19	the	the	DET
st74cn73540	7	20	second	second	ADJ
st74cn73540	7	21	project	project	NOUN
st74cn73540	7	22	we	we	PRON
st74cn73540	7	23	ask	ask	VERB
st74cn73540	7	24	for	for	ADP
st74cn73540	7	25	fixed	fix	VERB
st74cn73540	7	26	d	d	NOUN
st74cn73540	7	27	,	,	PUNCT
st74cn73540	7	28	m	m	PROPN
st74cn73540	7	29	,	,	PUNCT
st74cn73540	7	30	and	and	CCONJ
st74cn73540	7	31	δ	δ	PROPN
st74cn73540	7	32	,	,	PUNCT
st74cn73540	7	33	as	as	SCONJ
st74cn73540	7	34	the	the	DET
st74cn73540	7	35	g_i	g_i	NOUN
st74cn73540	7	36	vary	vary	VERB
st74cn73540	7	37	what	what	PRON
st74cn73540	7	38	is	be	AUX
st74cn73540	7	39	the	the	DET
st74cn73540	7	40	minimal	minimal	ADJ
st74cn73540	7	41	multiplicity	multiplicity	NOUN
st74cn73540	7	42	of	of	ADP
st74cn73540	7	43	f(i	f(i	PROPN
st74cn73540	7	44	)	)	PUNCT
st74cn73540	7	45	?	?	PUNCT
st74cn73540	8	1	this	this	DET
st74cn73540	8	2	study	study	NOUN
st74cn73540	8	3	leads	lead	VERB
st74cn73540	8	4	us	we	PRON
st74cn73540	8	5	to	to	PART
st74cn73540	8	6	investigate	investigate	VERB
st74cn73540	8	7	the	the	DET
st74cn73540	8	8	case	case	NOUN
st74cn73540	8	9	when	when	SCONJ
st74cn73540	8	10	i	i	PRON
st74cn73540	8	11	is	be	AUX
st74cn73540	8	12	a	a	DET
st74cn73540	8	13	strongly	strongly	ADV
st74cn73540	8	14	stable	stable	ADJ
st74cn73540	8	15	ideal	ideal	NOUN
st74cn73540	8	16	.	.	PUNCT
st74cn73540	9	1	we	we	PRON
st74cn73540	9	2	give	give	VERB
st74cn73540	9	3	classification	classification	NOUN
st74cn73540	9	4	results	result	NOUN
st74cn73540	9	5	for	for	ADP
st74cn73540	9	6	minimal	minimal	ADJ
st74cn73540	9	7	multiplicity	multiplicity	NOUN
st74cn73540	9	8	strongly	strongly	ADV
st74cn73540	9	9	stable	stable	ADJ
st74cn73540	9	10	elements	element	NOUN
st74cn73540	9	11	and	and	CCONJ
st74cn73540	9	12	also	also	ADV
st74cn73540	9	13	study	study	VERB
st74cn73540	9	14	when	when	SCONJ
st74cn73540	9	15	an	an	DET
st74cn73540	9	16	equigenerated	equigenerate	VERB
st74cn73540	9	17	2	2	NUM
st74cn73540	9	18	-	-	PUNCT
st74cn73540	9	19	borel	borel	PROPN
st74cn73540	9	20	ideal	ideal	NOUN
st74cn73540	9	21	is	be	AUX
st74cn73540	9	22	cohen-macaulay.finally	cohen-macaulay.finally	ADV
st74cn73540	9	23	,	,	PUNCT
st74cn73540	9	24	in	in	ADP
st74cn73540	9	25	the	the	DET
st74cn73540	9	26	third	third	ADJ
st74cn73540	9	27	project	project	NOUN
st74cn73540	9	28	,	,	PUNCT
st74cn73540	9	29	we	we	PRON
st74cn73540	9	30	investigate	investigate	VERB
st74cn73540	9	31	the	the	DET
st74cn73540	9	32	case	case	NOUN
st74cn73540	9	33	where	where	SCONJ
st74cn73540	9	34	i	i	PRON
st74cn73540	9	35	defines	define	VERB
st74cn73540	9	36	a	a	DET
st74cn73540	9	37	gorenstein	gorenstein	ADJ
st74cn73540	9	38	-	-	PUNCT
st74cn73540	9	39	linear	linear	NOUN
st74cn73540	9	40	ideal	ideal	NOUN
st74cn73540	9	41	for	for	ADP
st74cn73540	9	42	d	d	PROPN
st74cn73540	9	43	≥	≥	PROPN
st74cn73540	9	44	4	4	NUM
st74cn73540	9	45	.	.	PUNCT
st74cn73540	10	1	in	in	ADP
st74cn73540	10	2	this	this	DET
st74cn73540	10	3	case	case	NOUN
st74cn73540	10	4	,	,	PUNCT
st74cn73540	10	5	we	we	PRON
st74cn73540	10	6	are	be	AUX
st74cn73540	10	7	most	most	ADV
st74cn73540	10	8	interested	interested	ADJ
st74cn73540	10	9	in	in	ADP
st74cn73540	10	10	the	the	DET
st74cn73540	10	11	defining	define	VERB
st74cn73540	10	12	equations	equation	NOUN
st74cn73540	10	13	of	of	ADP
st74cn73540	10	14	the	the	DET
st74cn73540	10	15	special	special	ADJ
st74cn73540	10	16	fiber	fiber	NOUN
st74cn73540	10	17	ring	ring	NOUN
st74cn73540	10	18	.	.	PUNCT
st74cn73540	11	1	we	we	PRON
st74cn73540	11	2	describe	describe	VERB
st74cn73540	11	3	the	the	DET
st74cn73540	11	4	symmetric	symmetric	ADJ
st74cn73540	11	5	gorenstein	gorenstein	ADJ
st74cn73540	11	6	-	-	PUNCT
st74cn73540	11	7	linear	linear	NOUN
st74cn73540	11	8	ideals	ideal	NOUN
st74cn73540	11	9	when	when	SCONJ
st74cn73540	11	10	δ	δ	NOUN
st74cn73540	11	11	=	=	SYM
st74cn73540	11	12	3	3	NUM
st74cn73540	11	13	and	and	CCONJ
st74cn73540	11	14	use	use	VERB
st74cn73540	11	15	sagbi	sagbi	PROPN
st74cn73540	11	16	basis	basis	NOUN
st74cn73540	11	17	techniques	technique	NOUN
st74cn73540	11	18	to	to	PART
st74cn73540	11	19	exhibit	exhibit	VERB
st74cn73540	11	20	some	some	DET
st74cn73540	11	21	evidence	evidence	NOUN
st74cn73540	11	22	for	for	ADP
st74cn73540	11	23	a	a	DET
st74cn73540	11	24	conjecture	conjecture	NOUN
st74cn73540	11	25	.	.	PUNCT