Let’s start by discussing the major ancient sources that talk about the rape / abduction of Persephone:  Hymn 2 to Demeter (~700 BC), book five of Ovid’s Metamorphoses (~8 AD), and Claudian’s epic poem De raptu Proserpinae (~395-7 AD). There are a few differences between them, but they generally tell the same story.
In Hymn to Demeter, after Persephone is abuducted by Hades and trapped in the Underworld, Demeter (her mother) is devastated and travels the world, searching for her. Finally, as a last-ditch effort to get her daughter back, she (as the goddess of agriculture and vegetation) prevents any plants from growing, leading to widespread famine and starvation. Due to the impending threat of the complete annihilation of the human race, Zeus (Persephone's father and the king of the gods) is forced to take action. He confronts Hades (his brother and lord of the Underworld) and asks him to let Persephone return to the surface. Before this happens, though, Persephone is forced into eating some pomegranate seeds which, due to some strange law, means she cannot leave. Eventually, it's decided, as a compromise, that Persephone can come home but only for two-thirds of the year – the remainder, Demeter is plunged into unbearable grief once again, thus creating the seasons. 
Ovid’s Metamorphoses is a bit more embellished and dramatic, but continues in the same basic way. Ceres (aka Demeter) still traverses the earth trying to find her daughter, then appeals to Jove (aka Zeus) to have her returned. This time, however, Proserpina (aka Persephone) eats the pomegranate seeds herself, although she didn't understand the consequences of her actions. It still ends the same, with Persephone being returned but only for part of the year.
Claudian's epic poem, on the other hand, stops abruptly, with Ceres (Demeter) still searching in vain for her daughter, who remains trapped in the Underworld indefinitely. It's assumed that he just died before he could finish it, though.
As mentioned in the introduction, Bruce Lincoln explains in his article, "The Rape of Persephone: A Greek Scenario of Women's Initiation," that Persephone's story is likely meant to reflect the life of the average Greek woman at the time (223). As women weren't considered full citizens (lacking the right to own property, vote, or otherwise meaningfully participate in society), their lives were largely decided by the men around them (Cartwright). For example, a girl's father was in charge of arranging her marriage – neither she nor her mother had any say in the matter. When it was time, the husband-to-be would simply abduct her and do with her as he pleased (Lincoln 226-7). Unsurprisingly, this is also the case with Persephone. In the Hymn, Helios (the all-seeing god of the sun) tells Demeter, when asked what happened to her daughter:
“Queen Demeter, daughter of rich-haired Rhea, I will tell you the truth; for I greatly reverence and pity you in your grief for your trim-ankled daughter. None other of the deathless gods is to blame, but only cloud-gathering Zeus who gave her to Hades, her father's brother, to be called his buxom wife. And Hades seized her and took her loudly crying in his chariot down to his realm of mist and gloom. Yet, goddess, cease your loud lament and keep not vain anger unrelentingly: Aidoneus, the Ruler of Many, is no unfitting husband among the deathless gods for your child, being your own brother and born of the same stock: also, for honor, he has that third share which he received when division was made at the first, and is appointed lord of those among whom he dwells.” (lines 75-87)
Zeus planned for Hades to abduct Persephone, Helios explains, and he emphasizes that Hades would be a decent husband because of his high status as King of the Underworld. Again, all that mattered in regards to marriage in Ancient Greece was status and continuing the man’s bloodline — love had nothing to do with it, unlike how we in the West tend to see marriage today. The best a woman could really hope for was to eventually develop a sort of friendship with her husband, who was free to seek passion or pleasure outside of the relationship with whoever he wanted, while she tended to the household and the kids (Cartwright).
Another interesting thing to note is the symbolism of the pomegranate and its seeds. Lincoln says in Ancient Greek literature, pomegranates were often used to denote blood – not only of war, but also menstruation, loss of virginity, and childbirth. Fruits in general also often evoke images of fertility and sexuality, with the word for "seeds" being, both in Ancient Greek and modern English, a common euphemism for testicles or sperm (234).
Keeping this in mind, before her abduction in the Hymn, Persephone is only referred to as a maiden or Demeter's daughter (with occasional descriptions of her being "trim-ankled" or "deep-bosomed," emphasizing her beauty). It's only afterwards, when she's in the Underworld, that she's called Persephone. Lincoln suggests that this is because she is, in fact, no longer a maiden, so this placeholder name no longer applies. He further explains that it's a common custom across many cultures for young women to change their name after they've completed their rites of passage, namely loss of their virginity (229). In Western culture, for example, women are typically expected to take their husband's last name after marriage. While we lack proper records of daily life in the ancient world, this may suggest that something similar occurred in Ancient Greece, where a young woman would only receive a personal name after marriage (230).
Under this interpretation, its not just for some arbitrary reason that Persephone is confined to the Underworld, but specifically because she is no longer "innocent" – the marriage is seen as "consummated," and thus there's nothing to be done to change it.
I think it's so important to keep the actual historical context of the myth in mind while reading about it or any adaptations. It's like what I said earlier, people only seem to care about the fake Ancient Greece and Rome that exist only in their heads, discarding anything that interrupts their rose-tinted fantasies informed by what they personally want to see or various sanitized versions made up to the more "child-friendly." But ignoring reality only does a diservice to everyone: ancient, modern, and every time in between! We need to look at what little evidence we have and try to understand what life really was like back then and think critically about the conclusions they made, "their aesthetic choices," and how people were treated back then, rather than just blindly romanticizing it all and trying to replicate ancient aspects today without understanding why they came about.
This is a common subject of discourse about Persephone, so I might as well talk about it now. In English, her myth is commonly referred to as "The Rape of Persephone," while, in Latin, the word used is raptus, which would more closely translate to "abduction" rather than our modern concept of "rape." Some argue that this means that she never was raped and, as you'll see as you read the other chapters, this is used to justify many "alternative interpretations" of her story, but I want to clarify that this alleged attempt to avoid historical revisionism, is itself a form of historical revisionism.
For example, in Claudian's epic poem, Persephone says:
O fortunate are those girls whom other ravishers
Have borne off! At least they have delight in the usual daylight.
But along with my virginity, the sky is taken from me;
My purity is snatched away with the light, and I must depart from earth
As I am led captive into slavery for the Stygian tyrant. (Lincoln 228)
Similarly, in Hymn to Demeter, it states:
She was sitting in bed with Hades, her bedmate,
Much against her will, and yearning for her mother. (Lincoln 228)
Yes, the modern idea of consent did not exist in Ancient Greece (mostly because women were literally seen as property and not as actual human beings with rights), but I still think these examples make it clear that she was forced by Hades to have sex against her will – ie: what we would now refer to as rape. Just because this is a difficult topic to discuss doesn't mean we should completely brush over it or try to pretend it doesn't exist! Just because a practice is common in a culture doesn’t mean it isn’t harmful, sexist, or otherwise beyond reproach! ➥
This was a super condensed version of the story! If you were interested, a more in-depth overview of the classical sources and the differences between them can be found here. Otherwise, Hymn to Demeter is fairly short, interesting, and surprisingly readable. ➥
Anonymous. The Homeric Hymns and Homerica with an English Translation by Hugh G. Evelyn-White. "Hymn 2 to Demeter." Cambridge, MA., Harvard University Press; London, William Heinemann Ltd. 1914. Online.
Cartwright, Mark. "Women in Ancient Greece." World History Encyclopedia, 2016. Online.
Lincoln, Bruce. "The Rape of Persephone: A Greek Scenario of Women's Initiation." The Harvard Theological Review, Vol. 72, No. 3/4 (Jul. - Oct., 1979), pp. 223-235. Online.